Shamos’s Catalog of the Real Numbers
by
Michael Ian Shamos, Ph.D., J.D.
School of Computer Science
Carnegie Mellon University
2011
Copyright © 2011 Michael Ian Shamos.
All rights reserved.
PREFACE
i
Preface
Have you ever looked at the first few digits the decimal expansion of a number and tried to
recognize which number it was? You probably do it all the time without realizing it. When
you see 3.14159... or 2.7182818... you have no trouble spotting  and e, respectively. But are
2  3
and
you on familiar terms with 0.572467033... or 0.405465108...? These are
12
log 3  log 2 , both of which have interesting stories to tell. You will find them, along with
those of more than 10,000 other numbers, listed in this book. But why would someone bother
to compile a listing of such facts, which may seem on the surface to be a compendium of
numerical trivia?
2  3
Unless you had spent some time with
, you might not realize that it is the sum of two
12


cos k
very different series,  ( 1) k 2 and  k  (2k )   (2k  1)  . The proof that each sum
k
k 1
k 1
2
 3
is elementary; what connection there may be between the two series remains
equals
12
elusive. The number log 3  log 2 turns up as the sum of several series and the value of certain

1
,

k
k 1 3 k
( 1) k 1
,

2k k
k 1



dx
dx
1 ( x  1)( x  2) and 0 2e x  1 . If your research led to
either number, this information might well lead you to additional findings or help you explain
what you had already discovered.
Mathematics is an adventure, and a very personal one. Its practitioners are all exploring
the same territory, but without a roadmap they all follow different paths. Unfortunately, the
mathematical literature dwells primarily on final results, the destinations, if you will, rather
than the paths the explorers took in arriving at them. We are presented with an orderly
sequence of definitions, lemmas, theorems and corollaries, but are often left mystified as to
how they were discovered in the first place.
The real empirical process of mathematics, that is, the laborious trek through false
conjectures and blind alleys and taking wrong turns, is a messy business, ill-suited to the crisp
format of referred journals. Possibly the unattractiveness of the topic accounts for the scarcity
of writing on the subject. How many unsuccessful models of computation did Turing invent
before he hit on the Turing machine? How did he even generate the models? How did he
realize that the Halting Problem was interesting to consider, let alone imagine that it was
unsolvable? What did his scrap paper look like?
This book is a field guide to the real numbers, similar in many ways to a naturalist’s
handbook. When a birdwatcher spots what he thinks may be a rare species, he notes some of
its characteristics, such as color or shape of beak, and looks it up in a field guide to verify his
identification. Likewise, when you see part of a number (its initial decimal digits), you should
be able to look it up here and find out more about it. The book is just a lexicographic list of
10,000 numbers arranged by initial digits of their decimal expansions, along with expressions
whose values share the same initial digits.
definite integrals:
ii
PREFACE
This book was inspired by Neil Sloane’s A Handbook of Integer Sequences. That work,
first published in 1971, is an elaborately-compiled list of the initial terms of over 2000 integer
sequences arranged in lexicographic order. When one is confronted with the first few term of
an unfamiliar sequence, such as {1, 2, 5, 16, 65, 326, 1957,...}, a glance at Sloane reveals that
this sequence, number 589, is the total number of permutations of all subsets of n objects, that
n
n
n n
n!
1
 n! , which is the closest integer to n ! e . Sloane’s book is
is,  k!   
k 1  k 
k 1 ( n  k )!
k 1 j!
extremely useful for finding relationships among combinatorial problems. It leads to
discoveries, which help in turn generate conjectures that hopefully lead to theorems.
What impelled me to compile this catalog was an investigation into Riemann’s -function
that I began in 1990. It struck me as incredible that we have landed men on the moon but still

1
have not found a closed-form expression for (3)   3 . I say incredible because in the 18th
k 1 k

1
century Euler proved that ( s )   s is a rational multiple of  s for all even s, and gave a
k 1 k
formula for the coefficient in terms of Bernoulli numbers. No such formula is known for any
odd s. It is not even known whether a closed form exists.



1
1
( 1) k
,
and
, whose values
I began exploring such sums as  s


s
st
st
s
st
k 2 k  k
k 2 k  k
k 1 k  k
can often be written as simple expressions involving various values of the -function. For


 2 1   coth 
1
5
1
,
and the remarkable
example,  5



(
)
3



3
4
2
4
6
2
k 1 k  k
k 2 k  k

1
1
. To keep track of these many results, I kept them in a list by numerical


4
2
12
k 2 k  k
value, conveniently obtained using Mathematica®. Very quickly, after experimenting with

sums of the form
 ( ak  b)  1 ,
I began to notice connections that led to the following
k 1
theorem.
For a  1 an integer, b an integer such that a  b  2 and c a positive real

1
( aj  b )

number, then  a  b
.

b
k
cj
k 1 ck
j 1
A proof is given at the end of the chapter. The proof is elementary, but I never would have
been motivated to write down the theorem in the first place without studying the emerging list
of expressions sorted by numerical value.
After a time, I became somewhat obsessed with expanding the list of values and
expressions and found that as it grew it became more and more useful. I began to add material
not directly related to my research and found that it raised more questions than it answered. It
became for me both a handbook and a research manifesto.
Theorem 1.

iii
PREFACE

For example, when I observed that the sums
1
and

2k
k 1 2

(2 k )
4
k 1
k
digit prefix, this evoked considerable interest, since sums of the form
1
shared the same 20-
1
a
k 0
bk
are very difficult
to treat analytically. After verifying that the sums were equal to over 500 digits, I began to


1
1
(2 k )
 ( 3k )
and  k
, to
look for generalizations, exploring such sums as  3k ,  4 k ,  k
k 0 2
k 0 4
k 1 2  1
k 1 4  1
give a few examples. The results suggested a conjecture, which soon yielded the following
theorem:

Theorem 2.
For c > 1 real and p prime,
i 1

1
c
p
i
 
k 1
 ( kp )
c kp  1
, where  ( n ) is the
Moebius function.
For a proof, see the end of the chapter. Alternatively, the theorem may be expressed in the


 (kp)
1
form  k
 
 p k  . This allows us to relate such strange expressions as
k 1 a  1
k 1 a 1 / p 


k 1

1
 
2
k
and

1
 e
k 1 3
3k
to Moebius function sums. Unfortunately, the theorem seems to


1
1
and
.
k


2
kk
k 1 k
k 1 k
This book is what is known in computer science as a hash table, a structure for indexing in
a small space a potentially huge number of objects. It is similar to the drawers in the card
catalog of a library. Imagine that a library only has 26 card drawers and that it places catalog
cards in each draw without sorting them. To find a book by Taylor, you must go to drawer “T”
and look through all of the cards. If the library has only a hundred books and the authors’
names are reasonably distributed over the alphabet, you can expect to have to look at only two
or three cards to find the book you want (if the library has it). If the library has a million
books, this method would be impractical because you would have to leaf through more than
20,000 cards on the average. That’s because about 40,000 cards will “hash” to the same
drawer. However, if the library had more card drawers, let’s say 17,576 of them labeled
“AAA” through “ZZZ,” you would have a much easier time since each drawer would only
have about 60 cards and you would expect to leaf through 30 of them to find a book the library
holds (all 60 to verify that the book is not there).
Hashing was originally used to create small lookup tables for lengthy data elements. The
term “hashing” in this context means cutting up into small pieces. Suppose that a company
has 300 employees and used social security numbers (which have nine digits) to identify them.
A table large enough to accommodate all possible social security numbers would need 109
entries, too large even to store on most computer hard disks. Such a table seems wasteful
anyway, since only a few hundred locations are going to be occupied. One might try using just
the first three digits of the number, which reduces the space required to just a thousand
locations. A drawback is that the records for several employees would have to be stored at that
location (if their social security numbers began with the same three digits). An attempt to store
shed no light on trickier sums like
iv
PREFACE
a record in a location where one already exists is called a “collision” and must be resolved
using other methods. There will not be many collisions in a sparse table unless the numbers
tend to cluster strongly (as the initial digits of social security numbers do). Using this scheme
in a small community where workers have lived all their lives might result in everyone hashing
to the same location because of the geographical scheme used to assign social security
numbers. A better method would be to use the three least-significant digits or to construct a
“hash function,” one that is designed to spread the records evenly over the storage locations. A
suitable hash function for social security numbers might be to square them and use the middle
three digits. (I haven’t tried this.)
Getting back to the library, notice that any two cards in a drawer may be identical (for
multiple copies of the same book) or different. The only thing they are guaranteed to have in
common is the first three letters of the author’s last name. Now imagine that the library sets up
2620 drawers. Aside from the fact that you might have to travel for some time to get to a
drawer, once you arrive there you will probably find very few cards in it, and the chances are
excellent that if there is more than one card, you are dealing with the same author or two
authors who have identical names. Of course, it’s still possible for two authors to have
different names that share the same 20 first letters.
This book is a hash table for the real numbers. It has 1020 drawers, but to save paper only
the nonempty drawers are shown. But it differs very significantly from the card catalog of a
library, in which the indexing terms (the “keys”) have finite length. When you examine two
catalog cards, you can tell immediately whether the authors have the same name, even if their
names are very long. When you look up an entry in this book, you can’t tell immediately
whether the expressions given are identically equal or not. That’s because their decimal
expansions are of infinite length and all you know is that they share the same first 20 digits.
With that warning, I can tell you there is no case in this book known to the author in which two
listed quantities have the same initial 20 digits but are known not to be equal. There are many
cases, however, in which the author has no idea whether two expressions for the same entry are
equal, but therein lies the opportunity for much research by author and reader alike.


sin k
sin 2 k   1
As a final example, seeing with some surprise that 
 2 
led me to an
k
k
2
k 1
k 1
investigation of the conditions under which the sum of an infinite series equals the sum of the
squares of its individual terms. (We may call such series “element-squaring.”) Elementsquaring series rarely occur naturally but are in fact highly abundant:

Theorem 3.
Every convergent series of reals B   bk such that B
k 1

(2)

  bk 2   can
k 1
be transformed into an element-squaring real series B   bk by prepending to B a single real
k 0
1
term b0 iff B( 2 )  B  . Proof: see the end of the chapter.
4
I hope that these examples show that the list offers both information and challenges to the
reader. It is a reference work, but also a storehouse of questions. Anyone who is fascinated by
mathematics will be able to scan through the pages here and wonder if, how and why the
entries for a given number are related.
Eventually it became clear that perusing the catalog in pursuit of gems by eyeball would
not be effective. I then wrote software to search for relationships among the entries, seeking,
v
PREFACE
for example, numbers p, q, r and s such that p = f (q, r, s) for interesting functions f. With
10,000 entries, such searches are fruitful. With a million entries more computing power would
be needed. Some of the results of these investigations can be found in the author’s papers, e.g.,
Overcounting Functions1 and Property Enumerators and a Partial Sum Theorem2. I contend
that this method of search yields a rich stream of conjectures, which in turn lead to interesting
mathematical investigations.
The selection of entries in this book necessarily reflects the interests and prejudices of the
author. This will explain the relatively large number of expressions involving the -function.
For this I make no apology; there are more interesting real numbers than would fit in any book,
even listing only their first 20 digits. The reader is encouraged to develop his own
supplements to this catalog by using Mathematica to develop a list of numbers useful in his or
her own field of study.
Professor George Hardy once related the now-famous story of his visit to an ailing
Ramanujan in which Ramanujan asked him the number of the taxicab that brought him. Hardy
thought the number, 1729, was a dull one, but Ramanujan instantly countered that it was the
smallest integer that can be written as the sum of two cubes in two different ways, as 13  12 3
and 93  10 3 . Hardy marveled that Ramanujan seemed to be on intimate terms with the
integers and regarded them as his personal friends3. It is hoped that this book will help make
the real numbers the reader’s personal friends.
Proofs of Theorems 1-3
For a  1 an integer, b an integer such that a  b  2 and c a positive real

1

number, then 
. A proof is given at the end of the chapter.

j
ab
 kb
c
j 1
k 1 ck
 
1
Proof: The left-hand sum may be written as the double sum   aj  b j . Reversing the
c
j 1 k 1 k
Theorem 1.

( aj  b )

order of summation,

 k
j 1 k 1

1
aj  b
c
j
1
b
k 1 k



1
1



aj j
b
j 1 k c
k 1 k


j 1
1
ck 
a
j
, which equals


1
1
1
. QED.



b
a
ab
 kb
k 1 k ck  1
k 1 ck

1
 ( kp )


, where  ( n ) is the Moebius
i


kp
p
1
i 1 c
k 1 c

Theorem 2. For c > 1 real and p prime,
function.

Proof:
 ( kp )
c




 ( kp )

1
c
appears at least once in
pi
 1 k 1 j 1 c
i 1
the double sum, certainly in the case when k  1 and j  p i . We will show that the sum of all
k 1
1
kp
jkp
. Note that every term of
Available at http://euro.ecom.cmu.edu/people/faculty/mshamos/Overcounting.pdf
Available at http://euro.ecom.cmu.edu/people/faculty/mshamos/PST.pdf
3This incident is partially related in Kanigel, R. The Man Who Knew Infinity. New York: Washington Square Press
(1991). ISBN 0-671-75061-5. The book is a fascinating biography of Srinivasa Ramanujan.
2
vi
PREFACE
1
in the double sum is zero except when k  1 and j is a power of p, in which
i
cp
case  ( kp)  1, and the result follows.
Suppose that jk is a power of p. Then if k  1,  ( kp)  0 since kp has a repeated factor.
Therefore, only the terms with k  1 contribute to the sum and  ( kp)  1 each j a power of p.
Now consider all pairs j, k with jk  i not a power of p. Suppose that i has the factorization
a1
p1  pq aq , with each of the ai  0 . If p k or if k has any repeated prime factors, then
 ( kp)  0 . Since k i , it suffices to consider only those k whose factors are products of subsets
Pk of distinct primes taken from the set P  p1  p q  p  , a set having cardinality q  1 or q,
coefficients of
depending on whether P contains p. If Pk has an odd number of elements, then  ( kp)  1;
otherwise, if Pk has an even number of elements, then  ( kp)  1.  ( kp)  0 cannot be zero
since p does not divide k. But the number of subsets of Pk having an odd number of elements
is the same as the number of subsets having an even number of elements. Therefore, the terms
cancel each other and terms with jk not a power of p do not contribute to the double sum.
QED.
Theorem 3.


k 1
k 1
Every convergent series of reals B   bk such that B( 2 )   bk 2   can

be transformed into an element-squaring series B   bk by appending to B a single real term
b0 iff B

(2)
Proof:
b
k
k 0
2
k 0
1
B .
4

For
B
to
be
 b0 2   bk 2  b0 2  B( 2 )  b0  B .
element-squaring,
it
is
necessary
that
Solving this quadratic equation for b0 yields
k 1
1  1  4 ( B ( 2 )  B)
1
, which is real iff B( 2)  B  . Since b0 is finite, B  b0  B
4
2
1
converges. It is also elementary to show that if B( 2)  B  , then there is no real element that
4
can be prepended to B to make it element-squaring. QED.
b0 
Note that B   does not necessarily imply that B
(2)
( 1) k
  . For example, let bk 
.
k

1
1 1
(2)
Then B    ,  , which is finite, but B   , which diverges. Likewise, B(2)   does
2 2
k 1 k
2
1

but B diverges.
not imply that B   . If bk  , then B( 2 ) 
k
6
USING THIS BOOK
vii
Using This Book
This book is a catalog of almost 10,000 real numbers. A typical entry looks like:
.20273255405408219099... 
=
log 3  log 2
1
 arctanh 
2
5


( 1)k 1
1



2 k 1
(2 k  1) k 1 2 k 1 k
k 0 5

=
log xdx
 (2 x  1)
1
1

AS 4.5.64, J941, K148
2
 ( x) sin  x sin 5 x
dx
GR 1.513.7
x
0
Each entry begins with a real number at the left, with its integer and fractional parts
separated. Entries are aligned on the decimal point. Three dots are used to indicate either (1) a
non-terminating expansion, as above, or (2) a rational number whose period is greater than 18.
Three dots are always followed by the symbol , which denotes “approximately equal to.”
The center portion of an entry contains a list of expressions whose numerical values match
the portion of the decimal expansion shown. The entries in the list are separated by the equal
sign =; however, it is not to be inferred that the entries are identically equal, merely that their
decimal expansions share the same prefix to the precision shown.
There is no precise ordering to the expressions in an entry. In general, closed forms are
given first, followed by sums, products and then integrals. At the right margin next to
expression may appear one or more citations to references relating the expression to the
numerical value given or another expression in the entry. When more than one expression
appears on a line containing a citation, it means that the citation refers to at least one
expression on that line. See also, “Citations,” below.
=
Format of Numbers
Numbers are given to 19 decimal places in most cases. Where fewer than 15 digits are
given, no more are known to the author. Rational numbers are specified by repeating the
periodic portion two or more times, with the digits of the last repetition underlined. For
example, 1/22 would be written as .04504545, which is an abbreviation for the non-terminating
decimal .0454545454545045... . Since rational numbers can be specified exactly, the = symbol
rather than  is used to the right of the decimal. Where space does not permit a repetition of
the periodic part, it is given once only. If the period is longer than 18 decimals, the  sign is
used to denote that the decimal has not been written exactly. No negative entries appear.
Ordering of Entries
Entries are in lexicographic order by decimal part. Entries having the same fractional part
but different integer parts are listed in increasing magnitude by integer part. Rational numbers
are treated as if their decimal expansions were fully elaborated. For example, these entries
would appear in the following order:
viii
USING THIS BOOK
.505002034457717739...
.5050150501
.505015060150701508...
.5050
.505069928817260012...

=

=

Arrangement of Expressions
In general, terms in a multiplicative expression are listed in decreasing order of magnitude
except that constants, including such factors as ( 1) k appear at the left. For example, 4 k ! 2 k k 3
is used in preference to 4 k 3 2 k k !. Additive expressions are ordered to avoid initial minus
signs, where possible. For example, 3  2 k is used instead of 2 k  3. These principles are
applied separately in the numerators and denominators of fractions.
Citations
Abbreviations flushed against the right margin next to an expression denote a reference to
an equation, page or section of a reference work. Page numbers are prefixed by the letter p, as
in “p. 434.” A list of cited works appears at the end of the book.
Other Catalogs
Simon Plouffe maintains a server at the University of Quebec at Montreal that allows
searching among 215 million real numbers for equivalent symbolic expressions. See
http://pi.lacim.uqam.ca/. This book, by contrast, contains only about 10,000 entries. One
would expect, therefore, that almost all of the numbers in the book would also be found in
Plouffe’s index. To test this hypothesis, I had a student write a script to look up each of the
10,000 entries on Plouffe’s server to see how many it would find. The result was
approximately five percent. The reason for such a low degree of overlap, it appears, is that the
expressions in Plouffe’s index were primarily generated by computer from templates, which
the real numbers in this book were assembled for the most part from the mathematical
literature.
WARNING
While great effort has been expended to ensure accuracy, the numerical values and
expressions in this book have not been checked thoroughly and should not be relied upon
unless verified independently.
FIRST-DIGIT PHENOMENON
ix
The First-Digit Phenomenon
In 1881, Simon Newcomb observed that in books of logarithmic tables pages near the front
tended to receive more wear than the later pages, suggesting that the earlier pages were used
more frequently. That implies that the numbers whose logarithms were being looked up were
not uniformly distributed. In particular, numbers whose leading digit was a 1 occurred more
frequently than those beginning with 2 (or any other digit). Newcomb proposed that the
frequency of first digit n in a random list of numbers (assuming meaning can be ascribed to
such a concept), should be log10 (n+1) – log10 n. That is, the frequency of numbers beginning
with 1 ought to be about 30%, while only 4.6% of numbers should begin with 9.
This phenomenon was rediscovered by Frank Benford in 1938 and it is now referred to as
Benford’s law. Its generalization to numbers in base b is that the frequency of digit d is logb
(d+1) – logb d. This catalog (and Plouffe’s collection) provide experimental data concerning
the first digits of the fractional parts of lists of numbers. In fact, both lists exhibit the first digit
phenomenon very strongly. Because the fractional first digits range from 0 to 9 (rather than 1
to 9), the appropriate form of Benson’s law is to take b to be 11, not 10.
The bar chart below shows a plot of log10 (d+1) – log10 d (solid black bars) versus the
actual percentage of entries in the catalog whose fractional part begins with the digit d
(diagonal shading).
No explanation is offered to explain the deviation from the numerical predictions of
Benford’s law in this instance, but the shape of the decay as d increases is similar. Various
mathematical derivations of Benford’s law have been offered, but none are rigorous because of
the futility of attempting to define a random sample of numbers form an unbounded set. It is
somewhat easier to understand the law informally. When I explained Benford’s Law to the
late John Gaschnig, an artificial intelligence graduate student at Carnegie Mellon in the late
1970’s, he pondered it for a few minutes and then announced, “Yes, it’s because 1 is first.”
NOTATION
x
Notation
a( n )
greatest odd divisor of integer n.
Ai( x )
Airy function, solution to y   xy  0 .
b( x )
 xk .

AS 10.4.1
( 1)k
k 0
Bk
kth Bernoulli number, Bk  Bk ( 0 ) .
AS 23.1.2
xt

k
te
t
  Bk ( x ) .
t
e  1 k 0
k!
Bk ( x )
kth Bernoulli polynomial, defined by
bp(n)
ck
number of divisors of n that are primes or powers of primes.
AS 23.1.1
coefficient of x in the expansion of  ( x ) .
k
C( x )

(1) k  2 k x 4 k 1
1
.
C ( x)   cos  t 2 dt  
2k
20
k  0 ( 2k )!2 ( 4k  1)
Ci( x )
cosine integral, Ci ( x)    log x 
d (n)
number of divisors of n, d ( n )   0 ( n )
dn
nth derangement number, d1  0, d2  1, d3  2, d4  9 , nearest integer to n! / e . Riordan 3.5
e(n)
number of exponents in prime factorization of n.
E( x )
complete elliptic integral E ( x ) 
x
 /2

AS 7.3.1, 7.3.11

cos t  1
(1) k x 2 k



dt
log
x

.

0 t
k 1 ( 2k )! ( 2k )
x
AS 5.2.2, AS 5.2.16
1  x sin 2  d .
0
En
En ( x )
1
k
nth Euler number, Ek  2 Ek   .
2

2 e xt
tk
th
n Euler number, defined by t
  Ek ( x ) .
e  1 k 0
k!
AS 23.1.2
AS 23.1.1

Ei( x )
exponential integral Ei ( x )  

e  t dt
xk
.
x




log

 x t
k 1 k!k
Erfc( x )
(1) k x 2 k 1
2


 0
 k  0 k!(2k  1)

complimentary error function = 1  Erf ( x ) .
f ( n)
number of representations of n as an ordered product of factors
Fn
nth Fibonacci number, F1  F2  1; Fn  Fn1  Fn2
Erf ( x )
error function
2
x
t
 e dt 
2
2


 e x
k 0
AS 5.5.1
2
(1) k 2 k x 2 k 1
.
(2k  1)!!
AS 7.1.5
AS 7.1.2
xi
NOTATION
0 F1 (; a; x )
F ( a; b; x )
1 1
xk
, where a(k )  a( a  1)...( a  k ) .

k  0 k ! a( k )

x k a( k )

hypergeometric function
 k !b
Kummer confluent hypergeometric function
k 0
Wolfram A.3.9
,
(k )
where a(k )  a( a  1)...( a  k ) .

x a( k ) b( k )
k 0
k ! c( k )
F ( a; b; c; x ) hypergeometric function 
Wolfram A.3.9
k
2 1
, where a(k )  a( a  1)...( a  k ) .
Wolfram A.3.9
( 1)k
 0. 91596559417721901505...

2
k  0 (2 k  1)



1
1


1   
n
n 
(3k  1) 
k 1  (3k  1)

G
Catalan’s constant
Gn
gn



1
1


1   
n
n 
(3k  1) 
k 1  (3k  1)
gd ( z )
Gudermannian, gd ( z )  2 arctan e 
z
harmonic number: Hn 
Hn
n

2
1
k
k 1
H en
even harmonic sum: H
H on
odd harmonic sum: H
o
e
n
n


n
1
 2k
k 1
n
1
 2k  1
k 1
1
( 1) j 1  ( j 1) ( n  1)
  j  ( j ) 
, j 1
( j  1)!
k 1 k
n
H
( j)
n
H
generalized harmonic number:
( j)
n
hn
 (1) k
(1) k



n

(6k  5) n
k  0  (6 k  1)
hg( x )
Ramanujan’s generalization of the harmonic numbers, hg( x ) 
In ( x )
modified Bessel function of the first kind, which satisfies the differential equation
jn
Jn ( x )
k
K(x)




J314
x
 k(k  x )
k 1
z 2 y   zy   ( z 2  n 2 ) y  0 .



1
1

1   

n
n 
(3k  1) 
k 1  (3k  1)

( 1)k x n2k
Bessel function of the first kind 
n2k
k  0 k !( n  k )! 2
1  k2
 /2
1
complete elliptic integral of the first kind

0
Kn ( x )

1  x sin 2 
Wolfram A.3.9
J311
LY 6.579
d
modified Bessel function of the second kind, which satisfies the differential equation
xii
NOTATION
z 2 y   zy   ( z 2  n 2 ) y  0 .
Wolfram A.3.9
Ln
 log k log(k  x)  n log k

, when x is an integer n.



k
k  x  k 1 k
k 1 
“little” Fibonacci numbers: Ln  2 Ln 1  Ln  2 ; L0  L1  1 OR
Ln
Lucas numbers: Ln  Ln 1  Ln  2 ; L0  2; L1  1
li( x )
logarithmic integral li ( x)  
l( x )

x
dt
 log t
0
Lin ( x )
polylog function Lin ( x ) 
log x
p
natural logarithm log e x
a prime number.


xk

n
k 1 k
denotes a sum over all primes.
p
pd (n)
product of divisors pd ( n ) 
n
pf (n)
d
d |n
1
partial factorial pf ( n )  
k 0 k !
p
pfac(n)
product of primes not exceeding n, pfac( n ) 
pq(n)
product of quadratfrei divisors of n, pq ( n )  2
pr (n)
number of prime factors of n
prime(n)
nth prime
r (n)
4  ( d )
p n
pr ( n )
HW 17.9
d |n
s( n )
smallest prime factor of n
sd ( n )
sum of the divisors of n, sd ( n )  1 ( n )
S( x )
x

 t 2 
(1) k ( / 2) 2 k 4 k  3
dt  
.
S ( x)   sin 
x
2
(
2

1
)!
(
4

3
)
k
k
k
1



0
S1 ( n, m )
AS7.3.2, AS 7.3.13
Stirling number of the first kind, defined by the recurrence relation:
n
x ( x  1)...( x  n  1)   S1 ( n, m ) x m .
AS 24.1.3
m0
S2 ( n, m )
Stirling number of the first kind, S 2 ( n, m) 
Si ( x )
sine integral, Si ( x ) 
m
1 m
(1) m  k  k n .

m! k  0
k

sin t
(1) k x 2 k 1
.
dt


0 t
k  0 ( 2k  1)!( 2k  1)
AS 24.1.4
x
AS 5.2.1, AS 5.2.14
xiii
NOTATION
t k (n)
 k  1  ei 
 , tk (1)  1 , where n   pi ei
i
 k 1 
 
i
This is the number of ways of writing n as a product of k factors.
T (s)
1
k
k
s
, where k has an odd number of prime factors
u k (n)
number of unordered k-tuples a1 ..., ak  whose product is n.
w( x )
w( x )  e  x erfc(  ix )  
Wn

(ix ) k
k  0  ( k / 2  1)
k
k

 (1)
(1) 




n
(6k  5) n 
k  0  (6 k  1)
2
z (n)
number of factorizations of n.
z2 ( x )
z2 ( x ) 
( s)

f
( s )

 (( k )  1) x
k

k 2

( 1) k 1
 ( s)  
s
k  0 ( 2 k  1)
2
 x (    (1  x ))
1 x
n
1
n 
k 1 k

n


lim  f (k )  f ( x)dx 
n 
1
 k 1


1
Riemann zeta function  s
k1 k
Euler's constant
J313
  lim   log n  0.57721566490153286
AS 6.1.3
(  log k )m

ks
k 1

 ( m) ( s )
derivative of the Riemann zeta function
 p ( s)
Prime zeta function


k 1
1
prime(k ) s
 (k )
(n)
( 1) k 1
 (1  21 s )( s)

s
k
k 1

1
 ( s)  
 (1  2  s ) ( s)
s
(
k
)
2
1

k 0
Mangoldt lambda,  (k )  log k if k is a prime power; 0 otherwise
k
Moebius function,  (1)  1,  ( n )  ( 1) if n has exactly k distinct prime factors, 0 otherwise.
( n )
number of different prime factors of n.
( s)
 ( s)
( s) 

xiv
NOTATION
v1
 (s)
3i 2 
3i 2 
 (1  i 3 )

v1  (1  i 3 )

 2 
2




s
  ( s  1)  s / 2 ( s )
2
d
 k ( n)
sum of kth powers of divisors of n,

1 5
golden ratio,
 1. 61803397498948482...
2
k
d |n
( n)
Euler totient function, the number of positive integers less than and relatively prime to n.
 n (z )
1
Lin ( z )  Lin ( z )  
2
 ( n)
polygamma function
( x)
kth derivative of the polygamma function
( n)
z 2 k 1

n
k  0 ( 2k  1)

[Ramanujan] Berndt Ch. 9
n 1
(n) 
1
k 1 k
 ( n)     
 (k ) ( x ) 
d k ( x)
dx k
n
 ( k )
k 1


s
2
 ( s)
 ( s)  1  ( s  1) s / 2 ( s )
( n)
overcounting function  ( n )  -1 + the number of divisors of the g.c.d. of the exponents of the
prime factorization of n.
REFERENCES
xv
References
Citations given in the right margin of the number list refer to works in which more
information about a number or equation may be found. The form of citation is generally a word
or abbreviation followed by a reference giving a page, section or equation number. In the case
of journals, the citation is by volume and page. Following is an alphabetical listing of reference
abbreviations used in the text.
Andrews
AMM
AS
Berndt
CFG
CRC
Dingle
ex
GKP
GR
Edwards, H. M.
Henrici
Hirschman
HW
J
K
LY
Marsden
Melzak
MI
Andrews, L. Special Functions for Engineers and Applied Mathematicians.
New York: Macmillan.
American Mathematical Monthly
Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions.
New York: Dover (1964).
Berndt, B. C. Ramanujan’s Notebooks. New York: Springer-Verlag
(1985). ISBN 0-387-96110-0
Croft, H. T., Falconer, K. J. and Guy, R. K. Unsolved Problems in
Geometry. New York: Springer-Verlag (1991). ISBN 0-387-97506-3.
Handbook of Mathematical Tables. Cleveland: Chemical Rubber Co. (2nd
ed., 1964)
Dingle, R. B. Asymptotic Expansions: Their Derivation and Interpretation.
London: Academic Press (1973).
exercise
Graham, R. E., Knuth, D. E. and Patashnik, O. Concrete Mathematics.
Reading, MA: Addison-Wesley (1989). ISBN 0-201-14236-8.
Gradshteyn, I. S. and Ryzhik, I. M. Table of Integrals, Series and
Products. San Diego: Academic Press (1979). ISBN 1-12-294760-6.
Riemann’s Zeta Function. New York: Academic Press (1974). ISBN 012-232750-0.
Henrici, P. Applied and Computational Complex Analysis.
Hirschman Jr., I. I. Infinite Series. New York: Holt (1962). ISBN 0-0311040-8.
Hardy, G. H. and Wright, E. M. Theory of Numbers. Glasgow: Oxford
(4th ed., 1960)
Jolley, L. B. W. Summation of Series. New York: Dover (1961).
Knopp, K. Theory and Application of Infinite Series. New York: Dover
(1990).
Lyusternik, L. A. and Yanpol’skii, A. R. Mathematical Analysis:
Functions, Limits, Series and Continued Fractions. Oxford: Pergamon
(1965).
Marsden, J. E. Basic Complex Analysis. San Francisco: Freeman (1973).
ISBN 0-7167-0451-X.
Melzak, Z. A. Companion to Concrete Mathematics. New York: Wiley
(1973). ISBN 0-471-59338-9.
Mathematical Intelligencer
NOTATION
Riordan
Seaborn
Sloane
Titchmarsh
Vardi
Wilf
Wolfram
xvi
Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley
(1958).
Seaborn, J. B. Hypergeometric Functions and Their Applications. New
York: Springer-Verlag (1991). ISBN 0-387-97558-6.
Sloane, N. J. A. A Handbook of Integer Sequences. New York: Academic
Press (1973). ISBN 0-12-648550-X.
The Theory of the Riemann Zeta Function. London: Oxford (1951).
Vardi, I. Computational Recreations in Mathematica. Redwood City:
Addison-Wesley (1991). ISBN 0-201-52989-0
Wilf, H. S. generatingfunctionology. Boston: Academic Press (1989).
ISBN 0-12-751955-6.
Wolfram, S. Mathematica: A System for Doing Mathematics by Computer.
Reading, MA: Addison-Wesley (2nd ed., 1991). ISBN 0-201-51502-4.
ACKNOWLEDGMENTS
xvii
Acknowledgments
No book like this one has previously appeared, probably because of the monumental
difficulty of performing the required calculations even with computer assistance. However,
with the development of the Mathematica® program, a product of Wolfram Research, Inc., it
has been possible to obtain 20-digit expansions of most of the entries without undue difficulty.
The symbolic summation capability of that program has been invaluable in developing closedform expressions for many of the sums and products listed. Even though my involvement with
computers extends back more than 40 years, I have no hesitation in declaring that Mathematica
is the most important piece of mathematical software that has ever been written and my debt to
its authors is, paradoxically, incalculable.
The tendency to be fascinated with numbers, as opposed to other, abstract structures of
mathematics, is not uncommon among mathematicians, though to be afflicted to the same
degree as this author may be unusual. If so, it because of the role played by a collection of
people and books in my early life. My grandfather, Max Shamos, was a surveyor who, to the
end of his 93-year life, performed elaborate calculations manually using tables of seven-place
logarithms. The work never bored him — he took an infectious pride in obtaining precise
results. When calculators were developed (and, later, computers), he never used them, not out
of stubbornness or a refusal to embrace new technology, but because, like a cabinetmaker, he
needed to use his hands.
I read later in Courant and Robbins’ What is Mathematics, of the accomplishments of the
calculating prodigies like Johann Dase and of William Shanks, who devoted decades to a
manual determination of  to 707 decimal places (526 of them correct). At the time, I couldn’t
imagine why someone would do that, but I later understood that he did it because he just had to
know what lay out there. I expect that people will ask the same question about my compiling
this book, and the answer is the same. The challenge set by Shanks is still alive and well —
every year brings a story of a new calculation —  is now known to several billion digits.
My father, Morris Shamos, was an experimental physicist with a healthy knowledge of
mathematics. When I was in high school, I was puzzled by the formula for the moment of
2
inertia of a sphere, M  m r 2 . That is was proportional to the square of the radius made
5
sense enough, but I couldn’t understand the factor of 2/5. Dad set me straight on the road to
advanced calculus by showing me how to calculate the moment by evaluating a triple integral.
As a teenager during the 1960s, I worked for Jacob T. (“Jake,” later “Jack”) Schwartz at
the New York University Computer Center, programming the IBM 7094. (That machine,
which cost millions, had a main memory equivalent to about 125,000 bytes, about one-tenth of
the capacity of a single floppy disk that today can be purchased for a dollar.) NYU had a large
filing cabinet with punched-cards containing FORTRAN subroutines for calculating the values
of numerous functions, such as the double-precision (in the parlance of the day) arctangent.
By poring over these cards, I learned something of the trouble even a computer must go
through in its calculations of mathematical functions. It was this exposure that drew me to
field that later came to be called computer science.
For the past 25 years I have been an enthusiastic user of Neil Sloane’s A Handbook of
Integer Sequences (now co-authored with Simon Plouffe) in combinatorial research, so it is
ACKNOWLEDGMENTS
xviii
particularly pleasing to note that this book itself, in a sense, consists entirely of such
sequences.
My greatest debt, however, is to Stephen Wolfram for his creation of Mathematica, the
single greatest piece of software ever written. In developing this catalog, I used Mathematica
on and off over a 15-year period to explore symbolic sums and integrals and to compute most
of the constants that appear in this book to 20 decimal places.
ABOUT THE AUTHOR
xix
About the Author
Michael Ian Shamos was born in New York in 1947. He attended the Horace Mann School
and the Columbia University Science Honors Program during high school. He learned
computer programming at Columbia and the NYU Computer Center on the IBM 1401, 1620
and 7094 computers during the 1960s. He graduated from Princeton University in 1968 with
an A.B. degree in Physics and a minor in Mathematics. His senior thesis, written under the
guidance of Prof. John Wheeler, was devoted to the theory of gravitational radiation.
He joined IBM as an engineer in 1968, developing programming systems to operate
automated circuit testing equipment and obtained an M.A. from Vassar College in 1970,
working under Prof. Morton Tavel in the field of acoustics. In 1970, he joined the National
Cancer Institute to write software in support of the Institute’s research in cancer chemotherapy.
In 1970, he began graduate study at the Yale University Department of Computer Science
where he laid the foundations for the field of computational geometry and was awarded the
M.S., M.Phil. and Ph.D. degrees, the latter in 1978. In 1975, he joined the faculty of the
Computer Science Department at Carnegie Mellon University, doing research in combinatorics
and the analysis of algorithms.
In 1979, Dr. Shamos founded Unilogic, Ltd., a computer software company in Pittsburgh.
He obtained the J.D. degree from Duquesne University School of Law in 1981 and has
engaged in the practice of intellectual property law since then, dividing his time with Unilogic
until 1987. He practiced intellectual property law with the Webb Law Firm in Pittsburgh,
concentrating in computer-related matters, from 1990-1998. From 1980-2000, he has acted as
an examiner of computerized voting systems for the Secretary of the Commonwealth of
Pennsylvania and the Attorney General of Texas.
Dr. Shamos is now Distinguished Career Professor in the School of Computer Science at
Carnegie Mellon, where he directs a graduate program in eBusiness Technology.
Dr. Shamos is the author of four published books, Computational Geometry (1985), with F.
P. Preparata, Pool (1990), The Illustrated Encyclopedia of Pool, Billiards and Snooker (1993),
and Shooting Pool (1998). He serves as Curator of the Billiard Archive, a non-profit
organization devoted the preservation of the history of cue games, as Contributing Editor of
Billiards Digest magazine, and Chairman of the Statistics and Records Committee of the
Billiard Congress of America.
Dr. Shamos lives in the Shadyside neighborhood of Pittsburgh with his wife Julie. They
were married in 1973. Their son Alex is a student at the University of Arizona in Tucson.
Their daughter Josselyn Crane lives nearby with her husband Dan and their infant daughter
Harlow.
For more information, consult the Author’s web page:
http://euro.ecom.cmu.edu/shamos.html
( 1) k ( k  1)  ( 1) k 1 k 2


k!
k!
k 0
k 1
k k


( 1) 2 ( k  2)
( 1) k k




k!
k !!
k 0
k 1

 (k )

k
k 1

(k )

k
k 1 2  1
3

k 1 sin k
(

1
)

k
k 1

2
log x
0 x 2  1

.00000000000000000000
=
=
=
=
=
=

1 .000000000000000000000 =
1
=
=
=
=
k
 (k  1)!
2k k

k  0 ( k  2 )!

1

k  0 ( k  1)!  k !

 k!!

1




((k  2)!! k!! 
k  0  ( k  3)!

2k 4  k 3  1 
3k 2  3 k  1



6
5
4
3
k7  k
k 2
k 1 k  3k  3k  k


 2k 
1
(2 k  1)!!
 
  k

k  1  k  4 ( k  1)
k  1 ( 2 k )! ( k  1)
2
=
 2k 
4k  3
  4 k  2

(k  1) 2
k 0  k  2
=



 ( k ) log k
k 1
k

=
=
Fk 1
F F
k 2


k 1
J396
½
2
k 1

=
Berndt Ch. 24 (14.3)
1
 2k
k 1

=
 2k  k 2
k
(

1
)
  k

 k2
k 1

 k ( k  1)
k 1

=
GR 1.212
k
k 1
F4 k  2
8k
GR 0.245.4

=
Hk
 (k  1)(k  2)
k 1

=
 2k
k 2

=
Lk
3
k 1

=
Hk

 2k
2
k
  ( k )  1
k 2

=
=
( k )
k
k 2

 ( k ) log k

k
k 1

  (k )

=
 (k ) 
  k (k  1)  k  1 

k 2


=
Berndt Ch. 24, Eq. 14.3
1
, where S is the set of all non-trivial integer powers

  1
S
(Goldbach, 1729)


 
1  log n k
=  
k!
n  1 n! k  2

k 1 sin 2 k
=  ( 1)
k
k 1
k 1

( 1)
= 1 
k
k 1


dx
log x dx
=  2  
x
x2
1
1
JAMA 134, 129-140(1988)

=
 (x
0
1
=
2
J1065
dx
 1) 3/ 2
dx
 (1  x)
0
1  x2

=
x log(1  x )
dx
ex
0


=
sin x  dx

   cos x 


x  x
0
 /2
=

0
dx
(sin x  cos x) 2
Berndt I, p. 318


 
=

1
2 .000000000000000000000 =
=
=
arctan x
dx
x2
 (½,1,0) 

k


k
k 1 2
=
=
k!2k


k 1 ( 2k )!

k
 (k  1)!!
k 1
1  ( 2)  9 
1  ( 2)  7 
 1 
 1 
     ( 2)   
     ( 2)    
512 
216 
 6 
6
 8 
8

k
Prud. 5.1.25.31

2
2
k 1 ( k  1 / 4)

( k  1) Hk
Fk



k
k 
2
k 1
k 1 2

 ( 2k )  1
exp 
k
k 1

z (k )

2
k 1 k

=


4
k 0
k
 2k 
1
 
( k  1)  k 
  (2k )  (2k  1) 


2k
 1 22 k 1  1 
k 1

=
  2


1
1 
 

k (k  2) 
k 1 

1 

1 2 k 

2 
k 0 

=
=

=
 e(1)
k 1

=

0

2
cos x dx 
1
x
dx 
0

=

x11 dx
0

3 .000000000000000000000 =
ex
k  2k  1


k 
 k 
6
k 0
3
k 2  2k  1

k 2  2k
k 1

Prud. 6.2.3.1
1/ 2 k k
x  sin x
dx
1  cos x
 log sin x
e
e
x
o e x dx 
0
=

k2


2
k 2 k  1
k 1
 2
2
 /2
=
/k
log x
1 x 2 dx 
 /2
=
k 1

2
d
 (cos ) 
0
2 1
GR 3.791.1

=
=
Fk Lk
k
k 1 3


2 
1 


k (k  3) 
k 1 


 2 k  2
k ( k  1)
1







k 1
k
k 
2k
k 1 2
k 2
k  0 4 ( k  1) 


( 2 k )!!
( 2 k )!!




2
1
k  2 ( 2 k  1)!! ( k  k )
k 1 ( 2 k  1)!! k ( k  2 )

3 

1  2


k  4k 
k 1 


FF
k2
 (½,2,0)   k   k kk 3
3
k 1 2
k 1
2

k

k  3 ( k  2 )( k  2 )
 3

x dx
log 3 xdx
0 e x  1 x 2

( k  4 )  Fk k
 k

2k
k 0
k 1 2

4 .000000000000000000000 =
=

 
=
6 .000000000000000000000 =
=
=
10 .000000000000000000000 =
k

( k  1) 2


2k
k 0

12 .000000000000000000000 =


2

dx
 xe  x dx
x log x 0

15 .000000000000000000000 =
20 .000000000000000000000 =
24 .000000000000000000000 =
k4
(1 / 3, 4, 0 )   k
k 1 3
2

k

k  4 ( k  3)( k  3)

log 4 xdx
0 x 2

26 .000000000000000000000 =
( k  1)3

2k
k 0

k2

k  5 ( k  4 )( k  4 )

Fk k 2

k
k 1 2

k4
(½
, 4 , 0 )   k
k 1 2

52 .000000000000000000000 =
70 .000000000000000000000 =
94 .000000000000000000000 =
150 .000000000000000000000 =
k3
k
k 1 2
(½
, 3, 0 )  

1082 .000000000000000000000 =
k5
k
k 1 2
(½
, 5, 0 )  

Fk k 3

k
k 1 2

F k4
25102. 000000000000000000000 =  k k
k 1 2

1
1 .00000027557319224027...  

k 1 (10 k )!
1330 .000000000000000000000 =


1 .000002739583286472...

 .00000484813681109536... 
1 .000006974709035616233... 

 .00001067827922686153... 
1 .0000115515612717522...
1 .000014085447167...
1 .0000248015873015873...



648000
, the number of radians in one second
coth 2
 10

 (1) k 1
547  61 7
(1) k 1 





7

210395
(6k  1)7 
k 1  (6k  5)
 W7

1
  3
k 1 ( k )!
 .00003354680357208869... 
1 .00003380783857369614... 
=
 .00004539992976248485... 
1 .0000513451838437726...

J313

.00004439438838973293... 

697  8
 W8 
6613488

1 .00005516079486946347... 
J313
 9
sec1 
1
12 arctan 2  6 arctan 4  3  6 arctan 4  3  4 3  3
72

37  20 3  
1 
5

  3 log 3  log 343  3 3  2 arctan
 log

13
72 
3



dx
1
 11 23 1 
,1, ,

   12
2 F1 
x 1
22528  12 12 4096 
2


e 10 


 

511( 9) 
1
AS 23.2.20

9
512
k  0 ( 2 k  1)
1
 ( 5)  1 
 5
 7
 11 
      ( 5)     ( 5)     ( 5)   
 12 
 12 
 12 
 12  
358318080 
 (9) 
 ( 1) k 1
( 1) k 1 




6
(6k  1) 6 
k  1  ( 6k  5)

=
J328
1 .0000733495124908981...

W6 
91 6
87480

 .00010539039165349367... 
 8

 .00012340980408667955... 
e 9

6 .00014580284304486564... 
 ( 3) ( 2) 
1 .000155179025296119303... 
 (8 ) 

1 .0002024735580587153...

3
31


log 3 k


k2
k 1
17  8
255(8) 
1


8
161280
256
k  0 ( 2 k  1)
AS 23.2.20


.0002480158734938207...
J313
1
 (8k )!
k 1


1
1
 1
 (2)    
3 
 16 
8192
k  0 (16k  1)


1
.0002499644008724365...   12
   (6k  6)  1
 k 6 k 1
k 2 k

 (1) k 1
5  61 7
(1) k 1 


1 .0002572430074464511... 



5

2736
(6k  1)5 
k 1  (6 k  5)
1
1

.00026041666666666666 =
3840 5! 2 5
1 .000249304876753261748... 
.0002908882086657216...








, the number of radians in one minute of arc.
10800

1
 49 
.000291375291375291375 =
  1  2 
k 
3432 k  8 
 (3)
1
 .00029347092362294782... 
 
 ( 2 ) 1 
4096
8192
 .00033109368017756676... 
 7
 .00033546262790251184... 
e 8 
1 . 00038992014989212800... 
1 .000443474605655004...

W5 
J328
3751 (5)

3888



k 1
1
  (6k  1)
k 1

1
 (16k )
5

3

1

5 
(6k  5) 
J313
307  7
655360 2
 (1) k 1
(1) k 1 




7

(4k  1)7 
k 1  ( 4k  3)

(1) k / 2 

7
k  0 ( 2k  1)

=
=
J326
Prud. 5.1.4.3

1
1
3  1
1 .0004642141285359797... 
  
7776  6  k 1 (6k  1) 4
127( 7) 
1
1 .00047154865237655476...   ( 7) 

7
128
k  0 ( 2 k  1)

1
1 .00049418860411946456...  (11)   11 
k 1 k

1
1
 1
1 .00056102267483432351...  
 ( 2)    
3 
 12 
3456
k  1 (12 k  11)

1 .000588927172046...

r(16 ) 
AS 23.2.20
Berndt 8.14.1
5
log(1  2 )
2 2
2 2 2
2 2
2 2 2


log 2 
log
log
8
4 2
16
16
2 2 2
2 2 2

1
21
9
15
( 1) k 1 k 3
 .00064675959761549737...  1   ( 2 ) 
 ( 4 )   ( 3)   (5)  
5
2
2
2
2
k 1 ( k  2 )
 ( 3) 
1
.00069563478192106151... 

1728 k 1 (12 k ) 3
2

k  1 k ( 2 k )
1 .00076240287712890293...   ( 1)
2k
k 1




 .000807441557192450666... 
1 1 3
5 
   2 
64  2   

.00091188196555451621... 
e 7

0
x cos(2 x)
e 2
x
1
dx 
1
k
k 1
1 .00099457512781808534... 


1
1
 11 
 (2)    
 12  k 1 (12 k  1) 3
3456
.00088110828277842903... 
1 .00097663219262887431... 

5k

1
10
k 1 k
(10)  

.00099553002208977438... 
  (10k )  1 
k 1

.000998407381073899767... 
k
k 2


10
1

 k2

1
(1  i 
1  i 
  5    2 coth   1  i  2  cot
 

coth



6 16 
2
2






13
1
.00101009962219514182... 

  (4)    (6k  4)  1   10
12
 k4
k 1
k 2 k



1
( k )
.00102614809103260947...    (5k  5)  1   10



5
5
k
k 1
k 2 k
k 2 k
 
=
1
2

.001040161473295852296... 
 6 


15 
1
 coth    (2)   (6)    (4 k  6)  1   10
 k6
8 4
k 1
k 2 k

1
36 

.00108225108225108225 =
  1  2 
924 k  7 
k 

1
1
 5
.00114484089113910728...  
 (2)    
 6  k 1 (12 k  2) 3
3456
1  ( 3)  1 
 5
 7
 11  
1 .00115374373790910569... 
      ( 3 )     ( 3 )     ( 3)    
 12 
 12 
 12 
 12  
124416 
.0010598949016150109...

 ( 1) k 1
( 1) k 1 




4
(6k  1) 4 
k  1  ( 6k  5)


 
=

1
1
 1
 (2)    
 9  k 1 (9 k  8) 3
1458

2k k 2
2 
 (k  1)  1   
3 .00126030124568644264...  
3
2
k 1 k  1
1 .001242978120195538...

.001264489267349618680... 
1 .00130144245443407196... 

22
 
7
1
x 4 (1  x) 4
0 1  x 2 dx
1
 ( 5)  1 
 7
 3
 5
      ( 5)     ( 5)     ( 5)    
 8
 8
 8
 8
31457280 
 ( 1) k 1
( 1) k 1 




6
(4 k  1) 6 
k  1  ( 4 k  3)


 
=
197
3 ( 3)  ( 1) k



3
27  64
32
k  0 ( 2 k  8)
e 1
cos1 
1
.00138916447355203054... 
 1

4 4e
2
k 1 ( 4 k  2 )!
.001311794958417665...

3 .00139933814196238175... 
1 .0013996605974732513...

.00142506011172369955... 
.00144498563598179575... 
.00144687406599163371... 
1 .0014470766409421219...

 2 
3 3
log 3 log( 3  1)
log 2 

 r (12 ) 
6
4
3

1


10
log k
k 2 k

960
1
 ( 2k )
1 6  1
 


 ( 6) k  2 k 6

(2 k )


6
k 2 k  1
6
63 ( 6) 
1

  ( 6) 

6
960
64
k 1 ( 2 k  1)
Bendt 8.14.1
AS 23.2.20
=
1

1 
 ( 2k )
k6
k 2


1
1
 9
 (2)    
3
 16 
8192
k  0 (16 k  9)

log 4 k
(4)
24 .00148639373646157098...   ( 2 )  
k2
k 1

1
1
(2)  3
.00153432674083843739...  
  

 4  k 1 (12 k  3) 3
3456
 .00146293937923861579... 
.0015383430770987471...
.0015383956660985045...
1 .001712244255991878...



930 (5)   6
k


6
1920
k 1 ( 2 k  1)
4   ( 3)  ( 1) k



3
256
k  0 ( 4 k  8)

1
1
( 2)  1
 
  

 8  k 1 (8k  7) 3
1024

5 ( 3)
6
1
 1
1 .00181129167264869533... 
 ( 3)   
 4
1536
1 .00171408596632857117... 


1
 (4k  1)
k 0
4

1

 
=
1 log3 x
dx
6 0 x 4  1
5 2 197 
1

 2

2
54 216 k 1 k ( k  1) ( k  2 ) 2 ( k  3) 2

7 3  (3)
1
.001871372759366027379... 

  3 2k

360
2
 1
k 1 k  e
.001815222323088761...


=
 3 (k )
e 
1
1
.001871809161652456210...   2 2k

 1
k 1 k  e
k 1

2 k

.001872682449768546116... 
1
 k e 
2 k
k 1

.00187443047777494092... 
e 
k 1


 1


k 1
e 2k
 1 ( k )
k 1

1
2 ( k )
k 1

1
2 k

e 2k
 0 (k )
e 2k

.001877951661484336384... 
.001912334879124743...
x 6 dx
45 (7)
0 e2x  1  8 7
15e 2  109 
2k



960
k  0 ( k  6)!


LY 6.21
Ramanujan
1 .001949804343...

j9 
1 .00195748553...

G9 
.001984126984126984126 =
2 .001989185473323053259... 
 
=
J309
1

504
sinh 



J311
5

k 1
sin (1)

1 / 10
x5
e 
2 x
0
1
dx
 sin (1)3 / 10  sin (1)7 / 10  sin (1)9 / 10  
1 
10 

 1  k
k 1
1 .00200839282608221442... 
(1) k 1 k 5

k
 (1) k
 e

( 9) 


.00201221758463940331... 
.00201403420628152655... 
=
.00201605994409566126... 
.00201681670359358...

.00202379524407751486... 
.002037712107418497211... 
.00203946556435117678... 

 .00205429646536063477... 
.002083


1

1
k 1
k 2
3
3
1
5
9
17
387
 ( 6)   (5)   2 ( 3)   ( 4 )   ( 3)   ( 2) 

4
2
2
4
8
16
64

Hk 1


5
k 1 ( k  2 )


1
 (8k  1)  1   k 9  k

k 1
k 2
3

3277 
( 1) k



3375 32 k  0 ( 2 k  7) 3


1

(
7
k

2
)

1


  k9  k2

k 1
k 2
7
MHS (9,1)   (10)   2 (5)   (3) (7)
4


1
 (6k  3)  1   k 9  k 3 

k 1
k 2

7 (3)
1
1
 1
 (2)    
 
3
 2
4096
8192
k  0 (16k  8)
  (9k )  1   k
1

480

x 7 dx
0 e 2x  1

.00210716107047916356... 

1
  (5k  4)  1   k 9  k 4 
k 1
k 2

4
 ( 4k )  1
 

sinh 
k
90
k 1

1
1
(2)  2 
 
  

 3  k 1 (12 k  4) 3
3456
.002124728805039749548... 
.0021368161356763007...
9
1

4
 log


k 2


1 1
 4  
4 
 k 
  log1  k
13 
1
   (5)   2  i )  2  i  
8
2
4


1
=    ( 4 k  5)  1   9
5 
k 1
k 2 k  k
5 4

 W4 
486
.00213925209462515473... 

 
1 .0021511423251279551...



 2


log 4 2  ( 3)  log 2
19  5
 3 log 2 2




11520
96
48
8

 2k 
1
 
=  k

5 
k  0 4 ( 2 k  1)  k 

1
1
(2)  8 
.00227204396400239669...  
  

 9  k 1 (9 k  1) 3
1458
1 .002203585872759559997... 
.00228942996522361328... 
1
i 
 2  (1)1 / 3 )  2  (1)1 / 3  
  (3)   (6)   
6 2 3


1
 
=  9
( ( 3k  6)  1) 


6
k 1 k  k
k 1
.00234776738898358259... 

J313
1 .00237931966451004015... 
 ( 3)
1
512 k 1 (8 k ) 3
 /4
 log 2 G

   log sin x dx 
4
2
0
1/ 2
=




log x
dx
1  x2
2 log 2 log 5
3
1 5


log
 r (10) 
5
4
2
2 5

x 4e  x
3

 1  e 2x dx 
4 5

1109 
k2
17e 


24
k 1 ( k  5)!
e 6 

k 2 Hk
2
6 log 2  8 log 2  6   k

k 0 2 ( k  2)

x 4 dx
3 (5)

0 e2x  1
4 5
GR 4.224.2
GR 2.241.6
0
1 .0024250560555062466...

.002450822732290039104... 
.00245775047043566779... 
.002478752176666358...
2 .0025071947052832538...


.00254132611404785053... 

 .002597505554170387853... 

1 ( 2)  4 
  
432
 3
Berndt 8.14.1
1
1

284 4 ! 2 4
e
1
1
1
1
1
1
1 .00260426354837675067... 

 cos 
 cos  cosh  
2
2
2
4
2
2
4 e
.00260416161616161616


=
 .002608537189032796858... 


 .002613604652091533982... 
x
2
.00262472616135652810... 
.00264711657512301799... 

1
5 1
1
log  arctan 
8
3 4
4
x
7
2

1
 (4 k )!16
k 0
k
dx

 x 1
dx

1
7
log 2 

log 63

6
315 (9)

4 9

x
2
dx

x
7
8
x dx

1
e 
2 x
0
.00264953284061841586... 


1  
22 
22
6 
  log 31  2 10  2 5 arctan 13 
10
2
5
arccoth


 2 5  2 5    arctan 13 

20  
5 
5
5 



 
=
x
dx
7
2
1 .002651309852400201...

 x2
2 2

5

sin x  x cos x
dx
7/2
x
0

1
5 1
.00270640594149767080... 
log  
4
3 8
1 .00277832894710406108...  cosh 1  cos1

x
2
7
dx
 x3



log 7 1
dx
  7

3
24 2 x  x 4

1
 .00286715218149708931...   9
k  2 k log k

195353 3 (3)
(1) k 1

.00286935966734132299... 

 
3
216000
4
k 1 ( k  5)

 
.0028437975415075410...

log 2 
1 1 1
=
x5 y 5 z 5
0 0 0 1  xyz dx dy dz 

5 .0029142333248880669...


2k


k  0 k ! 1
 ( 3)
2
( 1) k 1 k

3
36
324
k 1 ( k  3)

1
1
 7
.00302826097746345169...  
 (2)    
3 
 16 
8192
k  0 (16 k  7)

1
1
(2)  7 
.0031244926731183736...  

  
 12  k 1 (12 k  5) 3
3450
.002928727553540092...

.00318531972162645303... 





2
e
3 2
sin 3 
sin 3x
x  2x  3


2
log 3
9
.00321603622589046372...  log 2 


2
24
.00326776364305338547... 
.003286019155251709...
.0033002236853241029...

Re (i ) 
1
7 4

16 45  256
3 (3)
7
.0033178346712119643... 


32
64
.0033104481564221302...
x
2
7
dx

 x5
5
1
 7
 
 ( 2)   
 9
1458



1
 (9 k  2)
k 1
3

(1) k


3
k  0 ( 2k  6)

1 1 1
x5 y 5 z 5
=   
dx dy dz 
2 2 2
0 0 0 1 x y z
24 .00333297476905227...
.0033697704469093...


.00337787815787790335... 
.003472222222222222222 =
.00364656750482608467... 

e(e 3  11e2  11e  1)
k4
1

4
0



,
,




e
 k 1 e k
(e  1)5

15 (5) 31
( 1) k


 
5
16
32
k  0 ( k  3)

1
1
(2)  7 

    
3
 8
1024
k  1 (8k  1)

1
k
 

288
k 1 ( k  1)( k  2 )( k  3)( k  4 )( k  5)

1
 ( 3)
( 1) k 1

 

3
27
36
K 1 ( 3k  3)
1 1 1
=
.00368755153479527...

x5 y 5 z 5
0 0 0 1  x3 y 3 z 3 dx dy dz
3
36 3

=

91 (3)
1 (2)  1
  
 
 6
216
432
1
 (6k  5)
k 1
3
Berndt 7.12.3
.0037145966378051377...

log(  log(  log(log 2 )))
 log(  log(log 2 ))
2
.00374187319732128820...  coth   1  2 

e 1
e  e
1 .00374187319732128820...  coth   
e  e

3
1
.00375000000000000000 =
 
800
k 1 ( k  2 )( k  3)( k  4 )( k  5)
5

19 
( 1) k 1
( 1) k 1
1 .00375568639565501099... 




5
( 4 k  1) 5
212 2
k 1 ( 4 k  3)

(1) k / 2 
= 
5
k 1 ( 2k  1)
1 .00372150430231012927... 
.0037878787878787878
1 .003795666917135695...
=

1

264
J 326
Prud. 5.1.4.3
(1) k 1 k 9

k
 (1) k
k 1 e

  4 arctan 3  2 log 5
2
8 2
dx
 1 x

4
0

32   3
( 1) k 1
 
3
256
k 1 ( 4 k  2 )

( 1) k k 3
22
2
.0038845985984860509... 
 2 log   k
7
3
k  0 2 ( k  1)
.00388173171751883...

1 .0038848218538872141...

arctan
1 .003893431348...

j8 
1 .0039081319092642885...


2
J 311
1
k
k 1

4k
656 8
1 .0039243189542013209... 
 G8 
6200145
1
.00396825396825396825 =
   ( 5) 
252

 1 
2 .00401700200153911987...   

9 

k 1  1  k


 
=
 



.00402556575171262686... 

J 309


25 
2

 1  k
k 6
1 /  (1)1 / 9 (1) 2 / 9  (1)1 / 3 (1) 4 / 9  
 1 /  (1)5 / 9 (1) 2 / 3  (1)7 / 9 (1)8 / 9 
7 (4) 7

 

8
8 4 sinh 
( 1) k

8
4 
k 2 k  k

(1) k (k )
(1)



4
k4
k 2
ω a non  trivial   1


 
=

integer power
.004031441804149936148... 
.00406079887448485317... 
.004061405366517830561... 

1
8 3
1

x 2 e x
0 1  e 2x dx 

9450

8

p 8 
 (8)  1

 (8)

 (k )

k 1
p prime
k
log (8k ) 
8
  (8)
9450
i

1 .00407802749637265757... 
sin (1)1 / 8 sin (1)3 / 8 sin (1)7 / 8 sinh (1)1 / 8  
4
2
1 .0040773561979443394...



 
=
 1  k 
8
k 2
15 
 2 sin  2  sinh  2

 coth  
16 8
8 cos 2  cosh  2


1
 
=   (8k )  1   8

k 1
k 2 k  1

16 3csch 
1
1 .00409337231139922783... 

 
8
cosh  2  cos  2
k 2 1  k
.00409269829928628731... 

.00410564209791538355... 
 10


1
   (7 k  1)  1

8
k 2 k  k
k 1

 3
3
tanh

.00413957343311865398... 
 ( 2) 
4
2
2 3


1
=  8

 (6k  2)  1

2
k 2 k  k
k 1
319 35 2
4
.004147210828822741353... 



512
768
576

1
 
=  2
5
k  2 ( k  1)
.00410818476208031502... 

2
1
arccos2 x
1 .00415141584497286361...  16 16 2  4 
dx
 
2
1 x
0
.004166666666666666666 =
1
   (7) 
240

x 3dx
0 e2 x  1 
MHS (8,1)  4 (9)   (2) (7)   (4) (5)   (3) (6)


1
  8
3    (5k  3)  1
k 2 k  k
k 1
15


  ( 4 )  coth  
8
4



( k )
1
=  8
4    ( 4 k  4)  1  
4
k 2 k  k
k 1
k 1 k
.004170242045482641209... 
.0042040099042187...
.0043397425545712...
.00442764193431403234... 
.00449344967863012707... 
1
.00450339052463230128... 
.00452050160996510208... 
3 .00452229793774208627... 
( 2)

74
( 1) k 1 k 2

 3 ( 3)  1  
4
2
180
k 1 ( k  2 )
1
1 ( 3)  1  
1
1 log 3 x

  
dx
4
486
6 0 x 3  1
 3  k  0 (3k  1)

(2 k )


k5
k 2

(2 k )

5
k 2 k  1
2
2
csc 2  2 
 2
4


 
=
1 .00452376279513961613... 
.004548315464154810203... 
.0045635776995554368...

.00460551389327864275... 

2k 2

2
2 
k 1
k  2 ( k  2)

31 (5)
1
  (5)  

5
32
k  0 ( 2 k  1)

3 2
1
1

 

2
2
2
256 9
k 1 ( 2 k  1) ( 2 k  1) ( 2 k  3)

G 3 3
( 1) k 1 k
1

 
3
2
64
k 1 ( 2 k  3)

dx
661
log 2 
  7
x  x6
960
2
 k 2 k  (2 k )  1 
.004641520347027523161... 

cot  2  2 

 

AS 23.2.20
LY 6.21

 1
2   (5)   (2)   1  i 3  2  (1)1 / 3  1  i 3  2  (1)1 / 3 
3 6


1

 
=  8
5    ( 3k  5)  1
k 2 k  k
k 1

1
1
 3
.00475611798059498829...  
 (2)    
3
 8
8192
k  0 (16k  6)
2 .0047586393290200803...

G
 log 2
2
1

log(1  x 2 )
0 1  x 2 dx 
GR 4.295.6
 /2

 
=

 x
 log sin 2  dx
0
2
log 1  2  r (8) 
4
67
2
4
( 3)
 (5)
.0047648219275670510... 





64
96
360
8
2

 (1) k 1
7 3
(1) k 1 

1 .00483304056527195013... 
  

3
216
(6k  1)3 
k 1  (6k  5)

7 ( 3)
1
.00486944347344743055... 
 
3
1728
k 1 (12 k  6)
1 .004759800630060566114... 
.00491550994358136550... 
.004983200...


log 2 
sin  2

2 4
mil/sec

 2
1  2 

k 
k 2 


1
 k ( k  2)
k 2
5
.00506504565493641652... 
.00507713440452621921... 
2 .005110575642302907627... 

 
=

1
1
(2)  2 

    
3
 3
1458
k  1 ( 9 k  3)

 4  93 (5)
k
 
5
192
k 1 ( 2 k  1)
4 10 3


3
81

k2
1
3 1
1 
5 1



2 F1  2,2, ,  
2 F1  3,3, ,  

2 4
3 
2 4
2
k 1  2 k 
 
k

.00513064033265867701... 
3 (3) 1549


4
1728
(1) k 1


3
k 1 ( k  4)

1 1 1

 
=
.005161189593421394431... 

.0051783527503297262...

.0053996374561862323...

 
=
.005491980326903542827... 
.005555555555555555555 =
6 .005561414313810886999... 
.00567775514336992633... 
x4 y4 z 4
0 0 0 1  xyz dx dy dz 


 (k )  1
13 
 coth  

30 8
2
16k
k 1

7 (3)
3
1
1
(2)  3
    

 
3
 4
128
512
1024
k  1 (8k  2)

15
  (2)   (4)   (6)    (2 k  6)  1 
4
k 1

1

8
6
k 2 k  k

13  4
x cos x / 2 
Prud. 2.5.38.10

dx
2

8
e 2 x  1
0

1
k
 
180
k 1 ( k  2 )( k  3)( k  4 )( k  5)
 2
2
x
  x dx
3
log 2
2
0
 (5, 3) 

1 log 2 log 3



1 .00570213235367585932... 
2
3
4

1
.005787328838611000493...   8

k 1 k log k
.005899759143515937...

45 ( 7 )

8 6

 216k
k 1
1
 6k 2
3
Ramanujan] Berndt Ch. 2
Berndt 9.27.12
1 .0059138090647037913...

1

1 .005912144457743732...
 
.005983183296406418...
250
1
 5
 ( 2)   
3 3

 (5k  4)
k 1
  (7) 

.00609098301750747937... 
log 2 x
 1 x
5
dx 
0
log k
7
k 2 k

1
 7
 (2)   
 15
3375


1


1
log 2 x
dx
x 8  x 8
log 2 x
dx
x8  x  7
1
 1
 
 (2)   
 2
196
2744
  (k )

   (k  1) 
1 .00615220919858899772... 
k 2


k
2 .00615265522741426944... 
k 1
2


1


1


1
log 2 x
dx
x 8  x 5
log 2 x
x8  x  4
5 2 2
1
2
 
arctan 2 
log
4
16
5 2 2

2
arctan 1  2  arctan 1  2

8




log 2 x
dx
x8  x  6
k
1
1
 7
.00624898534623674710...  
 (2)   
 12 
1728
.00625941279499622882... 

 (k  1) 

 (k ) 
1
 7
.00618459927831671541...  
 (2)   
 13
2197
31 .0062766802998201758...
2
1

 (3)
.00613294338346731778... 
8
3
1

1
 7
.00605652195492690338...  
 (2)   
 16 
4096


91 (3)
1 (1)  1 
1
 1

  
 ( 3)   
 6
 6  17496
243
486
324
2

( 2 k  1)
= 
4
k  0 ( 6 k  1)

 3 26
( 1) k


 

3
32 27
k  0 ( 2 k  5)

.00603351696087563779... 
 (1  2 )
1
3 
 log
2
0
x


log 3
2



8
16
x
dx 
(1  x ) x

x
2
6
dx

 x2
GR 4.261.10

=
.00628052611482317663... 
x 2e x / 2
 1  e x dx

2
48

log 2 3 ( 3)


8
32
GR 3.4.11.16
( 1) k 1 k 2


3
k 1 ( 2 k  2 )

5
log 21




12
2 3
3
3 3

1
dx
 5 11 1 
1
=
F
,
,
,


2 1
6

 6 6 64 
160
x 1
2

 .00629484324054357244... 

 
1
arctan

1
 7
.00633038459106079398...  
 (2)   
 11
1331
.00634973966291606023... 
1
log 2  log 31 
5

1

x
2
log 2 x
dx
x8  x  3
dx
x
6

4
k3
.006400000000000000000 =
  ( 1) k 1 k
625
4
k 1

1
log 2 x
(2)  7 
.00643499287419092255...  
     8
 10 
x  x 2
1000
1
.006476876667430481...

log 3 arctan 2 1 

 

4
2
2 4

x
2
6
dx
 x2
3
e e  4e  1
1
  k


,

3
,
0




k
(e  1) 4
 k 0 e
e
2
6 .006512796636760148...

102 .00656159509381775425... 
5
3
.0066004473706482057...
1 (2)  7 
 
   
 9
729
  (i )   ( i )
.0066201284733196607...

.006572038310503418...


1
log 2 x
dx
x 8  x 1
1
.00664014983663361581... 
log 2 1
log x
dx
 
4
6 0 ( x  1) 5

 ( 2 k  1)  1

k 1
.0067379469990854671...


 2 k 1
e 5
1 (2)  7 
.00675575631575580671...  
   
 8
512
log 2 x
1 x 8  1 dx

1
f2 (k )





4
4
i  2 j  2 ( jk )
k 1 k
3
1
 ( 6)   (5)   2 ( 3)   ( 4 )   ( 3)   ( 2 )  5
.00677885613384857755... 
4
2

H k 1
= 
5
k 1 ( k  1)

log 7
1
5
 3 1
dx
arctan
.0067875324087718381... 



  6
x  x3
6
6
8
3
3
2
.0067771148086586796...

 (4)  1
2




.00695061706903604783... 
1 .00698048496251558702... 
.00700907815253407747... 
1
11
2

arctan
108
8
2

 2
96
2
log 2  8 log 3 2  12 (3)  
2 (3)

343


4
k 0
k
 2k 
1

4 
(2 k  1)  k 
2
log x
dx
8
x
x
1

72
4
3 ( 3) 23
1
.00702160271746929417... 



 
4
48
180
4
8
k 1 k ( k  1)( k  2 )

 3 3
log 2 x
.00709355636667009771... 

G 2
dx
16 128
( x  1) 3
1
.00711283900899634188... 
.0071205588285576784...

9
3 1


8 e

6 .00714826080960966608... 
 ( 1) 
k 1
k 1

1 .007160669571457955175... 
k
k 0
.0071609986849739239...
.0071918833558263656...
( 1) k 1

k 1 ( k  4 )!

3
4
(2 k )  1 
1
2
1
54 (3)  2 2  45
24
 2 / 2
 e
 i i  cos( log i )  i sin( log i )

1 (2)  7 
.00737103069590539413...  
   
 6
216

x
1
8
dx
 x2
1
1 (2)  1
log 2 x dx
1 (2)  7 
2 .00737103069590539413...  
    2
     6
 6
 6
216
x 1
216
0

.007455925513314550565... 
 sech ( k )
2
k 1
.00747701837520388...
.00748486855363624861... 
.007511723574771330898... 
.0075757575757575757575 =
.00763947766738817903... 
1 .00776347048...

( 1) k 1 k
7  4 3 ( 3)

 
4
360
4
k 1 ( k  2 )

10016
( 1) k 1
G
 
2
11025
k 1 ( 2 k  7 )

1
1

  csch2 ( k ) 
6 2
k 1
1
   ( 9 )
132

log 3 13
dx

  6 4
2
24
x x
2
j7 
 1 

[Ramanujan] Berndt Ch. 26
J 311

1 .0078160724631199185...

1
 (k!)
k 1
7

28 4 2  6
1

  7
7
15
135
k 1 k ( k  1)

1
 

2
2
k 1 k ( k  1) ( k  2 ) ( k  3)
.0078161009852685688...
 1716  154  
.0078437109295513413...

119  2

72
6

G7 
1 .00788821...
2
J 309

2
1 (2)  7 
log x
dx
     8
 5
125
x  x3
1
3 (5)
.00798381145026862428... 
   ( 4 )   c4 
44

1
x 2e  x
.008062883608299872296... 

 1  e 2x dx 
4 3
.0079130289886268556...

.00808459936222125346... 


 


 1  log 2   
4 
2
x e
2  x2
log x dx
0

1
1
(2)  5 
.0081426980566204283...  
    
3 
 16 
8192
k  0 (16k  5)
1
1/ 8
2 .00815605449274531515... 
sin( ( 1) 3/ 8 sin( ( 1)5/ 8 sin( ( 1) 7 / 8  
4  sin( ( 1)



Berndt 9.27.12
 
=

1
8

 1  k
k 1
.00824553665010360245... 
1 
 1
 3 
 96 3  1536G   ( 3)     ( 3)   
 4
 4
6144 
.00824937559196606688... 
 (3)  13



k 2
f3 (k )

k3
1
 5
.0082776188463344292...  
 (2)   
 12 
3456
1
.00828014416155568957...  1 

 (7)
.008283832856133592535... 

p prime
1

p7

 (k )
k 1
k


Titchmarsh 1.2.14
1
 (12k  5)
k 0
3

log  (7k )

 3 
1 1 2 e
(1) k 1




cos 
   
.00830855678667219692... 

2 3e
3
3
k 1 (3k  2)!
 2

sin 1  sinh 1
sin 1 e
1
1
1 .00833608922584898177... 

 
 
2
2
4 4e
k  0 ( 4 k  1)!

1
1
k
 cos1  sin 1 
.00833884527896069154... 
 

8
8e
k  1 ( 4 k  1)!
1 .00834927738192282684... 
( 7) 

8 4 (3) 64 2 (5) 64 6 
 ( 2k )



k
1143
381
381 k  0 4 (2k  1)(2k  2)(2k  3)(2k  4)
=

=

=
1  p 1  p 2  p 3
 (4) 
1
 p  2  p 3  p  4  p 5  p 6
p prime 1  p
=
 (6)

6
(248k 3  2604k 2  9394k  11757) (2k )

7560 k  0 4k (2k  1)(2k  3)(2k  5)(2k  6)(2k  7)

Cvijović 1997
1  p 1  p 2  p 3  p 4  p 5

1
 p  2  p 3  p  4  p 5  p 6
p prime 1  p

.0083799853056448421...


  (8k  1)  1   k
k 1
.00838389135409569500... 
1

 k 1
MHS (7,4)  4 (5) (6)  21 (4) (7)  84 (2) (9) 
9 .00839798699900984124... 
 4
.00841127767185517548... 

.00847392921877574231... 
k 2
7

 ( 6k  1)  1
k 1
k
log(1  k 6 )
k
k 2

 
 1 i 3
1 2
1  1 i 3
 
  

  
4
3
3  2 
 2  



1
 
=  7
   (6k  1)  1
k 2 k  k
k 1

1
1
 5
 ( 2)    
.0085182264132904860...  
3
 9
1458
k 1 (9 k  4)


1
.00860324727442514399...   7
2    (5k  2)  1
k 2 k  k
k 1
8
  (3) (5)
7560

1
1
( 2)  5
.00866072400601531257...  
    
3
 8
1024
k  1 (8 k  3)
.008650529099561105501... 
1
1
2
 3
 7

  (2)      ( 2)   
 4  27
 4
64
64
7
 1
  (3)     i    ( i ) 
.00887608960241063354... 
8
2 4


1
 
=  7

 (3k  4)  1

4
k 2 k  k
k 1
.00877956993120154517... 

MHS (7,1) 


1
log 2 x
dx
x8  x 4
331
 (11) 
2
1 .00891931476945941517... 
 ( 6)

( 7)

5 .00898008076228346631... 
(
k 0
.00898329102112942789... 
.00915872712911285823... 
.009197703611557574338... 
.009358682075964994622... 
.0094497563422752927...



( k )
k7
k 1

Titchmarsh 1.2.13
1

k )!
2
e 3 / 2  i 3i

1 (2)  5
1
91 (3)
3

 
    
3 
 6
432
216
36 3
k  1 ( 6 k  1)

3
 

 
 ( 4k )  1

 coth   
 cot
9k
8 4 3
3
3
k 1
52 7
 
4e 2 4

 (1)k
k 0
3 ( 3)  log 2 
2
3
2k
k!(k  4)(k  7)
1 

 ( 1) k 1
k 1
k2
( k  2) 3
 3  i 3
5 2 
1   3  i 3
 
 


  
3
3
90 3   2 
 2  
4
.00948402884097088413... 



1

 (3k  4)  1


7
4
k 2 k  k
k 1

74
15
7 ( 4) 15
( 1) k



 
.00953282949724591758... 

4
45  16 16
8
16
k  0 ( k  3)
 16G   2   1  1  3  1 log 2 x dx
2 .00956571464674566328... 
      
64
 4
 4
1  x4
0
 
=
.00958490817198552203... 
.00960000000000000000
=
.00961645522527675428... 




96 2  2
6

625
 ( 3)



1
 (8k  7)(8k  5)(8k  3)(8k  1) 
k 1

log 3 x
1 x 6 dx

1
 ( 5k )
3
125
k 1
9
 (3)

.009632112894949285675... 

8
64
1 1 1
x5 y 5 z 5
 
= 
dx dy dz 
2 2 2
x
y
z
1

0 0 0
2 .00965179272267959834... 

7

108
3
2 .00966608113054390026... 

3 cosh  csc h 3 sinh  

2
log x
dx 
6
1
0
x


 1  k
k 1
2
1 

 3
J243
.009692044900729199735... 
153 .00984239264072663137... 
 (3)

4 3
5

x 2 dx
0 e2x  1
2

9( 3)   2
k
  ( 1) k 1
.00988445549319781198... 

96
(2 k  2) 3
k 1
1  ( 3)  1 
3
( 3)  3  
.0099992027408679778... 

        
 4
 4
3072 
64
.010007793923494035492... 
2 .0100283440867821521...


2
1
( 1) k
csch  3     2
6 k 0 k  3
3
i 2  (1)  i 
 i 
 i 
 i 
 
   (1)  
   2
   2
 
 2


 2 
4 
2
2

 
 

 
=

k
k 1
.010097259642724159142... 
 
2

  coth

2
2
Hk

 12
2

   (264k )  1
61 

1 2 
126 16
.01026128066531735403... 
3 (3) 1549


2
864
.01026598225468433519... 



1

k
k 1
log 2 x
dx
x 6  x5
4
1 (2)  7 
.01031008886672620565...  
   
 3
27

log 2 x
dx
x8  x5

1

69 3
 ( 3)
1


 
2
3
16
8
2
k 1 k ( k  1) ( k  2 )
7 (3)  3
1 ( 2)  1 
  
1 .01037296826200719010... 

 
 4
16
64
128

1  ( 2 )  1
1
 5 
 
= 
     ( 2 )     
3
 8 
 8
1024 
k  1 ( 4 k  3)
2
.01036989801169337524... 

.01037718999605679651... 

 (k )  1
k
k 2
.01041511660008006356...  1  cos
.01041666666666666666
=
1 .01044926723267323166... 
.010466588676332594722... 

 1
1
  Li  k   k 
6
k 2
1
1
cosh 
2
2

( 1) k 1
 (4 k )!4
k 1
k

GR 1.413.2

k2

k 1 ( k  1)( k  2 )( k  3)( k  4 )( k  5)

1
1 

4 sinh    1 

4
16 2 k 2 
k 1 
1

96
1
8

(2  3  2 log 2 ) (3)  2 (3)  
3


x 2 log x dx
0 e2x  1

7
1
log 2
2



log 1  2 
16
8
56
2
8


1
( k )  1
 
= 


2
8k
k  2 64 k  8 k
k 2

197
3 (3)
(1) k

 
.01049435966734132299... 

3
216
4
k  0 ( k  4)
.01049303297362745807... 

6


Berndt 7.12.3
cot
1 1 1
=
1 .01050894057394275299... 
x3 y 3 z 3
0 0 0 1  xyz dx dy dz
1  ( 3)  1 
 3
 5
 7
      ( 3)     ( 3)     ( 3 )    
 8
 8
 8
 8
24576 
 ( 1) k 1
( 1) k 1 
=  

4 
(4 k  1) 4 
k  1  ( 4 k  3)

 
(1) k / 2 

4
k  0 ( 2k  1)

=
3
5
9
81
2 (5)   ( 2 ) ( 3)   ( 4 )   ( 3)   ( 2 )  
2
4
8
16

Hk 1
= 
4
k 1 ( k  2 )
log 3 log 2
3
3 1  3 1


.01077586137283670648...  1 
log


8
4
12
3 1 8 3 1

1 
1
1  1
 (k )
 
=

hg

( 1) k





2
12 k 1 12 k  k 12  12 
12 k
k 1
.01065796470989861952... 


131
 .01085551389327864275...  log 2 

192
.010864219555727274797... 
6 .01086775003589217196... 
.01097098867094099842... 
.01101534169703578827... 

 

 .01105544825889466389... 
1 .0110826597832979188...


=

1 .011119930921598195267... 
 
=

x
2
1
 6 csc1  7  
12
6
dx

 x5
(1) k 1


4 4
2 2
k 1  k   k


1
1 ( 3)  1 

    6
4
256
 4
k  0 ( k  1)

1
log3 x dx
0 x 4  1
( 1) k

60
k  0 ( 2 k  1)( 2 k  2 )( 2 k  6)( 2 k  7 )

9
1
  ( 3)   (5)   7

5
4
k 2 k  k


1
   (2 k  5)  1 

5
k  1 k ( k  1) ( k  2 )
k 1

1  ( 3)  1 
( 1) k
( 3)  3  
1




 
 


4
 4
 4
1536 
k  0 ( 2 k  3)


1
 5 ( k )




k
5
2k
k 1 ( 2  1)k
k 1
1
cosh 2  cos 2 
2
 ( 1  i ) 2   (1  i ) 2 
1/ 4
3/ 4

 sin
 sin  1 sin  1   sin
2
2

 



49

3600




 
=

 1  
k 1
7 .01114252399629706133... 
1 

k4 
4
 3  12 log 2  9 log 3 

 (k 
k 0


1
1

1
2 )( k  3 )
1
2  cot1  coth1 
.01121802464158449834... 
4
csc 1
e2 cos1  cos1  3 sin 1  e2 sin 1 
 
=

2
4(e  1)

1
 
=  4 4
k 1 k   1

1 .01130477870987731055... 

   ( jk  3)  1
j  5 k 1

1
( 1) k k 3  2 k 
.01134023029066286157... 
 
 
8k  k 
36 6
k 0
log 3 2 log 2
1 .01140426470735171864... 

 r ( 6)
2
3
1 .01143540792841495861... 
32 log 2  8 ( 2 )  4 ( 3)  2 ( 4 )   (5) 


 
=

726 .01147971498443532465... 
1 .01151515992746256836... 
.0117335296527018915...


x6
0 e x (e x  1) dx
720( 7 )    (7) (1)



3 2 3 1


3
cos

12


dx
0 1  x12 

x10 dx
0 1  x12
1/ k
2
7
k 1 k



.01175544134736910981... 
2 .01182428891548746447... 
1
 k log
k 2

1

 k5

1  k2

  2
sinh 2 
2 k 1 k  4
1
cos(2 log x)
= 
dx 
2
(
1

)
x
0
.01174227447013291929... 
6
2k
720 (7)  1 

2 .01171825870300665613... 
 2k
k 1
 ( k  5)
k 1
.01147971498443532465... 

7
k
    ( s)  1 ds
 coth    

GR 3.883.1
s7

2
e 1
1 ( 2)  1 
   
125
5
2

1


3
k 1 (5k  4)
1
log 2 x dx
0 1  x5 
.0118259564890464719...

2


49

864
Hk
 k(k  1)(k  2)(k  3)(k  4)
144
k 1
2
3 ( 3) 
.01184674928339526517... 

 2 log 2  4 
2
2

 3 52
log 2 x
.01196636659281283504... 

  6
4 dx 
16 27

x
x
1
74
5 2
93



5760 192 256

e2  7
2k
.01215800309158281960... 
 
32
k  0 ( k  5)!


dx
.01227184630308512984... 
  2
( x  16) 2
256
0
.0121188016321911204...

( 1) k

2
4
k  2 ( k  1)


.01228702536616798183... 
csc1
2 sin 1  3 cos1 
6

.012307165328788035744... 
1 3 (3)


8
32
3 3
.01232267147533101011... 

1 .01232891528356646996... 
.01236617586807707787... 
.01243125524701590587... 
.0124515659374656125...

.01250606673761113456... 
8
1 1 1
3
( 1) k 1 k

3
k 1 ( k  1)( k  2 )

k 1
4
1
k  2k 2
4
x y z
dx dy dz
2 2 2
y z
  1 x
0 0 0

5 ( 2)  6 ( 3) 
 4  30 2
768

1

2

 (4 k
k 1
1

 1) 4
J373
2
18G  13 9 log 2
13



32 
32
128
80 2 8 4

 320 log 2  48 ( 3)  560 
3
45
1  4 ( 4)  8 ( 6)   ( 3)  12 (5) 


k
k 1
3
4
1
( 2 k  1) 4
k
 ( k  2)
k 1

J385
6

3 3  9 log 3
1
1 .01251960216027655609... 
 
 log 2  
8
2
8
k  0 ( 2 k  1)( 4 k  1)( 6k  1)

15
1
.01256429785698989747...  160  44 ( 2 ) 
 ( 4)  80 log 2  20 ( 3)   3
4
2
k 1 k ( 2 k  1)
4
5 2 163
.01261355010396303533... 



720
96
256

1
 
=  2
4 
k  2 ( k  1)
.012625017203357165027... 
4 2
1 7 (3)
 

9
6
2
Borwein-Devlin, p. 56
.012665147955292221430... 
.012706630570239252660... 
2 .01279324663285485357... 

1
8 2

xe  x
0 1  e 2x dx
15 (5)

4 5
 

e 
0
dx
2 x1 / 5
1



  kF!k! 
1
I0 5  1  I0 5  1 
5

k
k 1

1 .01284424247706555529... 
91 (3)
1  ( 2)  5 
 1 
     ( 2 )     W3

108
432 
 6 
6

log k
6
k 1 k

1
1
 3
.01286673993154335922... 
 ( 3)    
4
 4
1536
k  0 ( 4 k  3)
.01285216513179572508... 
   ( 6) 


31  3 2
1
 
2
108
k 1 k ( k  1)( k  2 )( k  3)

3 ( 3)
2
13
( 1) k 1
.0128978828974203426... 
 4 log 2 

 
3
2
4
24
4
k 1 k ( k  1) ( k  2 )

5 4
log 3 x dx
6 .01290685395076773064... 
  6
81
x 1
0
.01288135922899929762... 
.01292329471166987383... 
( 1) k 1


2
k 1 ( 2 k  5)

k 2  2k  2


2
k 1 k  2k  4
i
i 
log 3
1


1 .01295375333486096191...  arccot 2 
  log 3  i log1    i log1    
2
2
 2 
 2


1
dx
 
=  k
 2 2
x  x2
k  0 16 ( 4k  1)
2
.012928306560301890832... 

209
G 
225
 2 csch  3 sin  3 
.01303955989106413812...  1 
.01312244314988494075... 

 ( 1)
k 1
5 .01325654926200100483... 
1 .01330230301225999528... 
2
10
1  ( 2)  4 
(2)  5 
        
 3
 6 
216 

1 .01321183515...

k 1

log 2 x dx
1 x 6  x 3 
(k )
k2
8 
6
1
sin 
5  4 cos1 2

1
2
k 0
k
sin
2k  1
2
.01352550645521592538... 

7 (3) 7054
1
 7

   (2)   
 2
4
3375
8

1
log 2 x
dx
x8  x 6
1
( 1) k 1 k 3
2
3
3
12
2
2
  ( )  log       ( k  1) 3
4
k 0

49
1
.01361111111111111111 =
 
3600
k  0 ( 2 k  1)( 2 k  2 )( 2 k  6)( 2 k  7 )
2
4


k2
3
21
.01375102324650707998... 

  ( 3)  1  
4
32 128 16
k 1 ( 2 k  3)

65
1  4k  k 2
.01387346911505558557...   ( 2 )  6 ( 4 ) 
 

4
8
k  2 k ( k  1)

.01358311877719198759... 


 
=
 ( 1) k  ( k  2)  1
3
k
k 1
1 .01389384950459390246... 
.01389695950004687007... 
cos1  sin 1 
k 0

.01417763649649285569... 
.0142857142857142857
=
2 .01432273354831573658... 
5021 .01432933734541210554... 
3 .0143592174654142724...

2
coth    (4)   (2)  1 
2
 ( 1) k 1 (2 k  4)  1 



k
k 2
6
1
 k4
x
  ( x log
dx
1)( x  3)
log 3 2
  log 2 3 
12
2
0
3 ( 3)
( 1)
 

3
256
k 0 (4 k  k )

2 3
7
dx

  2
27
18
( x  x  1) 3
1

.01408660433390149553... 
1
k 1
k 1
1 .01406980328946130324... 

log 2 x
dx
x6  x2


=

  sink k

i
Li  e  i   Li2  ei  
2 2
 e /2
 e
 i ie
.014063214331492929178... 
1
 (4 k )!(2 k  1)
1  (2)  9 
( 2 )  5 
        
 8 
 8
512 
1 .01395913236076850429... 
.013983641297271329...

1

e
1

70

k

16 
2

 1  k
k 5

 1 (1 5 ) / 2
F
  k
e
5
k 1 k!
1
8
7
127
log x dx
 
240
1 x
0
e
5
1
e
5
5e
2 /(1 5 )


1
F
k 1
k
GR 4.266.1
.01442468283791513142... 
3 (3)

250

1 .0145709024296292091...

 e

log 2 x dx
1 x 6  x
( k ) 1
k 1
1 .01460720903672859326... 
.01462276787138248...

1 .01467803160419205455... 
.01476275322760796269... 
=
 1

 1  1  log 2 
kHk

e  1   Ei    
 1  
k

2 2  2
2 
k 1 k !2

(2 k )

4
k 2 k  1

7 ( 4 )
4
1

  ( 4)  

AS 23.2.20, J342
4
96
8
k 1 ( 2 k  1)
7 (3) 28


8
27
1 1 1
x4 y4 z 4
0 0 0 1  x 2 y 2 z 2 dx dy dz
1  ( 2 )  11 
(2)  5  
.01479302112711200034... 
        
 12  
 12 
1728 
2k  (k )
6 .01484567464494796828...  

k!
k 2

1 .014877322758252568429... 
1 .01503923463935248777... 


  e
2/k

k 1
.01521987685245300483... 
.01523087098933542997... 
.01529385672063108627... 
.01534419711274536931... 
.015439636746...

.01555356082097039944... 
log 2 x dx
1 x 6  1
2

 1

k

  H3

1
27
3 3
1
log
log 2  3 
4
8
4
e 
erf (1) 
4
2 k 1
k
k 1

e
 x2
sinh x cosh x dx
0
 3
4   
 4

17  2
4
 ( 3)
.01520739899850402833... 




16 48 180
4
1 .01508008428612328783... 

2
Berndt 8.17.16

k 1
( 4 )  3 ( 6)  3(5)   ( 7) 
2
1
 k ( k  2)

4

k3
 ( k  1)
k 1
7
( 2 k  1) 2

4
648
k  0 ( 6 k  3)

log 2
 1
( 1) k 1

  
8
16 8
k 1 ( 4 k  2 )( 4 k  4 )
2

 5
1
  ( 3)  
3
2
4
k 1 k ( k  1) ( k  2 )
J311
j6 
  6 1 i 3  3  i 3 1 i 3  3  i 3


 (3) 



3
6
6
 2 
 2 




 
=

 ( 1)  (3k  3)  1   k
k 1
k 1
k 2
6
1
 k3
45 (5)
 84 ( 3)  504 log 2  462 
.015603887018732408909... 
4
2 .01564147795560999654...   (8)

1
1 .01564643870362333537...  
6
k 1 ( k !)
.01564678558976431415... 
2 ( 6)  42 ( 4 )  252 ( 2 )  462 
1
1 .01567586490084791444...   3k
k 1 k
1
 1
.0157870776290938623...  
 (2)   
 4
8192

x

6
1
( k  1) 6
1
 (16k  4)
k 0
3

 (2 k )
k 1
2k

dx

1
5
2

.01593429224574911842... 
k
log 2
1 
2 1
2  1
 log
 
 
 log
2
2
2
2 

.01584647647919286276... 

k 1

1 .0158084056589008209...
( 1) k 1

6
6
k  1 k ( k  1)


k 1
 ( 4 k  1)
4 4 k 1  1
104  6
 G6
98415
1
243  32 3 3
1
 5
( 2)  1
.01594968819427129848...  
   

 ( 2)  


 3
3456
3
124416
3456

1
13 ( 3)
3


= 
3
1728
2592 3
k 1 (12 k  8)

2
2
log x
.01600000000000000000 =
 
dx
125
x6
1

1
1 .01609803521875068827...   k  4
k 1 k
1 .01594752966348281716... 
.01613463028439279292... 
.01614992628790568331... 
9

2
 ( 1)
k 1
.0161516179237856869...

x
2
2
 2 ( 3) 
3

=
log15

4
log 2 
10e 
k 1
dx
x
5

k
k 2
k 2  ( k  3)  1
163

6

k2
 ( k  4)!
k 1
2
k 1
( k  1) 3
J309
.0162406578502357031...

2 .01627911832363534251... 
1 .01635441905116282737... 
.0163897496007479593...

.016406142123916351060... 
31 (5) 7  2 ( 3)  4 log 2



8
32
48

2k

7
k 1 k ! k

1

5
k 1 k ! k

1  2G 
2
6

2
3

16
 log 2 
 sin  2  sinh  2 
2 2


Hk
 (2 k  1)
k 1
4
( 1) k

3
k  2 k ( 2 k  1)


1

2

1
k 1
 (4 k  2)  1
2   ( 1)

k
k 2
k 1
.0164149791532220206...  6   ( 2 )   ( 3)   ( 4 )   (5)   ( 6)



1
1


=  7
 ( k  6)  1


6
6
k 2 k  k
k  1 k ( k  1)
k 1
=
.01643437172288507812... 
.0166317317921115726...

.01666666666666666666
=
k
6

7 ( 3)
1
 
3
512
k 1 (8 k  4 )

( 1) k ( k )

k3
k 2
1

60
e 
0
dx
2 x1 / 4
1
 1  k 

2 .016707017649831720444... 

k 1
7

 1 / ((1)1 / 7 )((1) 2 / 7 )((1)3 / 7 )((1) 4 / 7 )((1)5 / 7 )((1)6 / 7 )

 (k )
.016733900585645006255...  
k 1 k!1
.01684918394299926361... 
log 7 1  3
3
5
arctan
 


6
2
6
3
3

.01686369315675441582... 
1


6
k 2 k  k
.01687709709059856862... 
6155  63
1080
2
29

12 36

nfac(k )
1 .016942819805640384240...  
2k
k 1
.01691147786855766268... 

4


x
2
 ( 1)  (5k  1)  1
k 1
k 1

log 3 x dx
  5
x  x4
1

1
 k ( k  1)( k  2)
k 1
2
( k  3)
5
dx
 x2

.0170474077354195802...
 1 
.017070086850636512954... 
945

6

1

p6

p prime
 ( 6)  1

 ( 6)

 (k )

k
k 1
log  (6k )
13 (3)
3
1 (2)  2 
.01709452908541040576... 
   

 
 3
216
432
324 3
.017100734033216426153... 

6
coth   1 


12
1 .01714084272965151097... 
1


6
k 2 k  1

3  2 2
8
6
2
2  2i 3 cot

=
 1 i 3

k 1
3
2i 3
2
 ( 1)  (6k )  1
k 1
k 1
   3
 2


e
dx
x2/3
0

1 .01729042976987861804... 
1
1 i 3

2
cot 


 ( 6k  2)
2
6 log 2     2

k 1 4 k  k
6
1

( k )
2
k 2
2 k 3
 ( 6)
945
1  log5 k
1  (5)
1 .01735897287127529182... 



  (m)
120 k  2 k ( k  1)
5! m 2

1
1 .01736033215192280239...   2

2
2
k 2 k  k

703 12
703
H (6) k
1 .01736613603473227207... 

 (12)   6
638512875 691
k
k 1
1 .01734306198444913971... 

1 .01737041750471453179... 
sinh 
cosh   cos  3 
4 3


  1  k1 
k 2
  (1  i ) 
3 
 (1  i )
1 i 
 coth   
  coth 
  2  cot 
16 8
2
2 
 16 

 


1
=  6
 2    (8k  2)  1
k 2 k  k
k 1
.01740454950499025083... 

 
6
.01745240643728351282... 
.01745329251994329577... 
.01745506492821758577... 
sin1

180
, the number of radians in one degree
tan1

  


.01746673680343934044... 
.01746739278296249906... 
.01758720202617971085... 
.01759302638532157621... 
.01759537266628388376... 
1 .01762083982618704928... 
.01765194199771948957... 
.017716230658780156731... 
.01772979515817195598... 

1

6
 1    ( 7 k  1)  1 
k 2 k  k
k 1


( 6k )  1
   log(1  k 6 )

k
k 1
k 2

33
k
 ( 3)  3 ( 4) 
 
4
16
k 1 ( k  3)


11

 3
1

tanh
  6
   (6k )  1
12 2 3
2
k 2 k  1
k 1
5


 (5k  1)  1
log(1  k )
 

k
k
k 1
k 2

 3 
 k6 
   6  
6 2 sech 2 



k 2  k  1
 2 

 2  8G
1
 
2
144
k 1 (12 k  3)

log(2 k  1)
15
4
Prud. 5.5.1.5
log 2   ( 4)  

4 
16
1440
k  0 ( 2 k  1)
1 11 log 2 log 3



6
18
4

e
1
x
dx
(e  1)(e x  2)(e x  3)
x
2
10G
1  (1)  1 
 5  
     (1)    


9
72 
 12 
 12   9
1  (1)  1 
 11 
 5
 7
=
     (1)     (1)     (1)    
 12  
 12 
 12 
 12 
144 
1 .017739549085798905616... 

 

 
 ( 1) k 1
( 1) k 1 




2
(6k  1) 2 
k  1  ( 6k  5)
1 2 3
 log 2  18 log 2 2 (2)  18 (2) log 2  6 (3) (2) 
12

(1) k 1 log3 k

k2
k 1


1
 (5k  1)  1



6
k 2 k  k
k 1

32 k 1
e 3  e 3
sinh 3 
 

AS 4.5.62
2
k  0 ( 2 k  1)!

( 1) k 1
1
10
 3 arctan  log
 1

k
3
9
k 1 2 k ( 2 k  1) 9

=
.017796701498469380675... 

 
=
.01785302516320311736... 
10 .01787492740990189897... 
.01793192101883973082... 

2 .01820187236017492412... 




  B1
1
 2  6 2 log 2 2  2 log 2  1 
72

k 1
2k
1 .01829649667637074039... 
 (5) 
.01831563888873418029... 
e 4
.018355928317494465988... 
3 (7)   (2) (5)   (3) (4)  MHS (6,1) 

Hk
 (k  1)
k 1
6


.018399137729688594828... 
  (2k )  (2k )  1) 
k 2

.018399139592798207065... 
  (2k )  (2k )  1)
k 2
.01840295688606415059... 

 .01840776945462769476... 

7
  (2)  coth  
8
4
3

512

 (x
0
2

1

6
2 
k 2 k  k

  (4k  2)  1
k 1
dx
 4) 3

Hk
1
 
54
k 1 ( k  1)( k  2)( k  3)( k  4)

7( 3)
4
k
.01856087933022647259... 

 
4
16
192
k 1 ( 2 k  1)
.0185185185185185185185 =

1 .01868112699866705508... 
 (k ) (k )  1
k 2

5
1
1
 3arctanh   k
3
2
k 1 4 ( 2 k  1)( 2 k  3)

 ( 3)
1
.01878213911186866071... 
 
3
64
k 1 ( 4 k )

dx
log 3 1
.01884103622589046372...  log 2 
   5

x  x3
2
8
2
.01874823366450212957... 

log 2 log 3
1

1




 
2
3
4
30 4 3
k  2 36 k  6 k

( k )  1
 
= 
6k
k 2
i 
1  1 
 i 
 1 

1 .01898180765636222260...    H  1 / 4   H  1 / 4   H   1 / 4   H   1 / 4  
4  2 
2 
 2 
 2 


 ( 4 k  1)
1
= 

  5
k
2
k 1
k 1 2 k  k

2 (3)
log 2 x
.01923291045055350857... 
dx
  6
125
x x
1
.01891895796178806334... 

1 .01925082490926758147... 
 (5)

 ( 6)


k 1
( k )
k6
Titchmarsh 1.2.13

3 .01925198919165474986... 
.019300234779277614553... 
.01931537584252291596... 
=
.019378701873...

e
dx
  x 4 1
e ( x  2)
2 2
0
1  (1)  7 
 9 
     (1)    
32 
8
 8 
k (2k  1)
64k
k 1


log 2 3
3
2 log 2 2
 1 1
2
3
log 
 log log 
 Li2   
 3 8
2
4
3
2

( 1) k

4
2
k
k 2 2 ( k  k )
1

3
k a non  trivial k
int eger power
.01938167868472179014... 
1
2
12

G


2 16
2
7
 .01942719099991587856... 

5 8
1
.019534640602786820175... 
219  10 2  72 (3) 
1728
.01939671967440508500... 

3 ( 3)

2
( 1) k 1 k

3
k 1 ( k  2 )

3
( 1) k 1 k 2

 
3
128
k 1 ( 2 k  1)

( 1) k ( 2 k  1)!!
 

( 2 k )! 4 k
k 2




  k (k  1)(kH 2H)(k  3(k  4)

k
k 1
k 1
.01956335398266840592... 
J 3 (1) 
.01959332075386166857... 
14
2
 1
 6 log  2 Li2    
 2
9
3
.01963249194967275299... 
 3  i 3
 3  i 3
4
1
  1 i 3  

  (3)   1  i 3  
3
6
 2 
 2  



=
=
1

6
3 
k 2 k  k
integer power
306 .01968478528145326274... 
.01972588517509853131... 
.01982323371113819152... 
5
3



  (3k  3)  1


k 1
1

ω a non  trivial   1

( 1) k 1 k 3

2
k
k  1 2 ( k  1)

(k )

3
k 1 k



1
2
( 1) k 1

 
2
9 108
k 1 ( 3k  3)

 4 17
1

  ( 4, 3)  
4
90 16
k 1 ( k  2 )
k
k 2
3

1
  (3)  1
1
.019860385419958982063... 
.0198626889385327809...

3 log 2 1
 
4
2

 64k
k 1

3 7 4


8 1920
x
1
5
3
1
 4k 2
[Ramanujan] Berndt Ch. 2
dx
 x3

3
3
3
k2H

log   ( 1) k 1 k k
64 32
4
3
k 1
log 5
1
1 5
log
1 .0199234266990522067... 

 r (5)
2
2
5


2 5
k2 1
k 2  (2 k )   (2 k  2) 
1 .0199340668482264365... 
   2


2
6 8
k  2 ( k  1)
k 1
1
.02000000000000000000 =
50

1
1 .02002080065254276947...  

k  0  2k 
3
  (2 k  1)
k
.01990480570764553805... 
.02005429455890484031... 
2 7 log 2


9
24

1  dx

1 log1  x  ( x  1)4 

x
dx
 log x  1  x
4
2
2
.02034431798198963504... 
.02040816326530612245... 
1 .02046858442680849893... 

 (2 k  1)!!
10 1
1
   

9 3
k  1  ( 2 k  1)!! 2 k  3
1
49
62 (5)

63

a(k )
6
k 1 k

Titchmarsh 1.2.13
.02047498888852122466... 
4 (5)  4( 4 )   ( 3)  1 
.020491954149337372830... 
 ( 3)
 ( 3) 

 (6)
1 .0206002693428741088...

J385
1
8
csc

9
9

x
0

k2

5
k 1 ( k  2 )
(1   ( k ) )

k3
k 1


1
4
n not squarefree n

dx
1
9

7 (3) 
1
 1

 2   2   (2)   
 4
8
32
64
3
.02074593652401438021... 
7 (3) 
1
 1

   ( 2)   
 4
8
32
64
3
2 .02074593652401438021... 
.020747041268399142...

e
5
 
2 e 1 4


Berndt 6.30

B2 k
 (2k )!4
k 1
k
1

1
2
log x
 1 x
0
log 2 x
dx
x6  x2
4
dx
1 .02078004443336310282... 
13 (3) 2 3
1
 1

   ( 2)   
 3
27
54
81 3

1
 (3k  2)
k 1
3
Berndt 7.12.3
1
=
33 .02078982774748560951... 
.020833333333333333333 =
1 log 2 x
dx
2 0 1  x 3
4
16

3
1
48

( k !) 2 3k
k  0 ( 2 k )!


1 
1 
1


1 .020837513725616828...
  2   ( 3)    1 
   1 
  5


2
2  k 1 2 k  k 3

 ( 2 k  3)
= 
2k
k 1

1
1
1 
1
1 .02083953399112117969... 
cos

cosh




k
4
4
2
2
2
k  0 ( 4 k )! 2

1
 

 
 (4 k )  1
.020839548935264637084... 

 coth
 cot
  
6 4 2
4k
2
2
k 1

( k )
3 .02087086732388650526...  
k 1 k !
3



log 2 x
csc  1  2 cot    5
2 .02093595742098418518... 
dx 
125
5
5
x 1
0

 
=

2
4 3 2 25  3 5

125 5  5

197 3 (3)
.02098871933468264598... 


108
2
1 .02100284076754382916... 

 
1  1 1
 ,3,  
8  2 2

log 2 x dx
1 x5  x 4
1
log 2 x
0 2  x 2 dx 
1   1 
 1 
 Li3 
  Li3  


2   2
2 
=
i   9  i 3  1
i   9  i 3
1
  

 
 
 
9  18 6 3   6   18 6 3   6 

1
 
= 
3
k 1 ( 3k  1)  1


 (k )  1
  1 1
.02108081020345922845...  

 Li5    

5
k k
k
k 2
k 2 
691
  ( 11)
.021092796092796092796 =
32760
.02103789210760432503... 


5/ 2

1
47

k (k  1)
1 (1)  2  
3
1 .021291803119572469996... 
  


2
3
3k
 3  k 1 (3k  1)
k 1
.02127659574468085106... 

.02129496275413122889... 

4k k
k 2
7
.02134847029391195214... 

.021491109059244877187... 

.021566684583513603674... 
 (k )  1

1 
 1

  
 log 1   
 4k  
k 1  4 k

3
csch 2 sin  3 
2

sech 
k2 1
  2

4
k 0 k  4
k 2  2k  2


2
k 1 k  2k  3


1 .021610099269294925320... 
3
3
3
arctan 4 x
log 2 
dx
 (3)  2 log 2 

4
6
12 0
x4
1 .02165124753198136641... 
2 log
.0217391304347826087...
5
1
 2
 4arctanh  2 Li1   
3
4
5
1

46
60 .02182669455857804485... 
1 .02210057524924977233... 
1
k 0
k

k 3 Hk
k
k 1 2

42  26 log 2 
2 2 arcsin

 16 (2k  1)
 2k 
1
 
k  0 32 ( 2k  1)  k 

1

2 2
k

7
log 3 x dx
.02219608194134170119... 
  5
30720
x x
1
1
.02222222222222222222 =
45

1
 2  ( 3) 29


 
.02222900171596697005... 
2
8
2
16
k 1 k ( k  1)( k  3)
 2 2  1
i  5i 2  1
i  5i 2
  

.02224289490044537014... 

 
 
 
 9 3 3  2   9 3 3  2 
54
9
4
i  (1)  5  i 2   1
i  (1)  5  i 2 
 1
 


 
 
 18 6 3 
 2   18 6 3 
 2 

 
  

 
=

 k
 1
1  (3)
.022436419216546791773... 


12 2 2
k 2
 
=
3
2

 ( k  1) (3k )  1
k 1
 ( 2k )

 4 (2k  2)(2k  3) 
k 1
k
1 
1
1 

 24k 2  1  48k 3arctanh
 12k 2 log1  2  

6 k 1 
2k
 4k  



1
.02258872223978123767... 
.02271572685331510314... 
.02272727272727272727
=
.0227426994406353720...

.02277922916008203587... 
.022891267882240749138... 

11
( 1) k 1 k
 
4
k 1 ( k  1)( k  2 )( k  3)
1
log 2
5
log x

 
4 dx

3
24
(
x
1
)
0
1
44


11
1
1
 ( 2)  ( 4)   6


4
4
4
k 2 k  k
k 1 k ( k  1) ( k  2 )
 /4
17 3
 log 2   sin 4 x tan 4 x dx 
16 2
0
4 log 2 
 3 (3) 
3


 
197
37 6
 2 (7) 11 4
 (9) 

  (5) 
  (3) 
24
2
120
7560
Hk
 

6
k 1 ( k  1)
Borwein-Devlin, p. 63
9  2 3 log 2 log 2 2




8 12
4
2

13
1
3
.02291843300216453709... 
log 3 
  3

8
2
k 2 9 k  k

1
1 .0229247413409167683...   4
k  2 k  15


1
 4 ( k )
1 .0229259639348026338...   k


4
2k
k 1 ( 2  1)k
k 1
.02289908811502851203... 
.02297330144608439494... 

 

1 .02307878236628305099... 
.02309570896612103381... 
1

(k  k 2 )
k 2

 ( 2 k  1)  1

9k
k 1
2
50G  43 28 log 2
71



128
128
1536



k
4
2
 (2 k  1)!!
1




k  1, k  3  ( 2 k )!!  2 k  6





1
2  2 sin  2 sec2

64
2 k 1 ((2 k  1) 2  2) 2
 log 
1

 1  log 2   ,  a zero of  ( s ) 
2
2
 


Edwards 3.8.4
31 (5) 7   ( 3) 7 ( 3)  log 2 





  2 log 2 
8
32
4
48
48 4

Hk
 
= 
4
k 1 ( 2 k  1)


log k
log 2
 ( 2k )  1


1 .02313872642793929553...   2

2
2k  1
k 2 k  1
k 1

log 2 
log 2
1
1
  H 2 k 1  (2k )  1 
   arctanh  
 
=
4
k
2
k 1
k 2  k
2
1 .02310387968177884861... 


4
4

5
( 1) k 1 k 3
 

216
5k
k 1



1
1
 2 (2  3 )

1 .02316291876308017646... 
  

2
2 
36
(12k  11) 
k 1  (12k  1)
.023148148148148148148 =


 
=
log x dx

x12  1
0


233
 coth 2 
1

  4
1920
32
k  3 k  16
1
.02325581395348837209... 
43

 2k 
1
 3  log 2 2

  k
1 .0233110122363703231... 

3 
48
4
k  1 4 ( 2 k  1)  k 
.02317871150152727739... 
1

 
=
arcsin x log x
dx 
x
0


.02334259570291055312... 

x 2 log x
dx
x
x
0 e  e  1
3  2  3 (3)  2 (3)  2  (3)   

(1) k 1
27
1 .02335074992098443049... 
 18 log 2  
2
2
k 1 k (k  1 / 9)

5 log 2
1
.02336435879556467065... 


  k
2
36
6
k 1 2 (9k  18k )

( 1)k
 
= 

k  0 (2 k  1)(2 k  2 )( 2 k  4 )(2 k  5)
.02343750000000000000
=
.02346705930540378299... 
3

128
( 1) k k 3


3k
k 1

2

sinh 2
K ex 107b

log 3 x
1 x5 dx
 k2 
 2


k 1  k  4 

26 (3)
4 3
1
1
5


   ( 2)  
27
27
81 3 4
 3





2 (3)
8 3
1
 4 3  i Li3  1  i 3   3  7i Li3   1  i 3  


 2 


3
2
81 3 6( 3  i ) 





2
log x
=
1 x6  x3 dx
.02351246536656649216... 
=


28567
.02353047298585523747...  2 ( 3) 
   ( 2 )  7 
12000




1
log 2 x
dx 
x8  x 7
.02375236632261848595... 

4


 1  log 2
 Li4   
21 (3)  log 3 2 
 2  24
24

S (k ,3)
 
=  1
k!k
k 3
1
.0238095238095238095
=
 B3
42

dx
 2 115  (1) k
.02385592231300210713... 


 6
2
5
12 144 k  0 (k  5)
1 x  x
1 .023884750384121991821... 
90

 2 log 2 2
2
1
cosh 
2
2
 1

1 .02394696928321706214... 


 1  
k 1
2
1

2
(2 k  1) 

   ( jk  2)  1
j  4 k 1

.02402666421487355693... 
k
k 2
6
1
   ( ( s)  1)dx
log k
( 1) k (2 k  1)
.02408451239117029266...  

2 k ( k 1) / 2
k 0


s6
3
1

1  k  

2 
k 1 
1 .02418158153589244247...  16 log 2  4 ( 2 )  2 ( 3)   ( 4 ) 


 
=

k 1

HW Thm. 357

 2k
k 1
( k  4)
5
1

 k4
2k
21/ k

6
k 1 k
1 (2)  1 
1
G 2
 1
1 .02434757508833937419... 


  
 ( 3)   


 4
8 64 256
4
6144
G 2
 3 7 (3)
1
 1




 
=
 ( 3)  
 4
8 64 128
32
6144
1
.02439024390243902439 =
41
2035
.02439486612255724836...   (3,5)   (3) 

1728
1 1 1 4 4 4
x y z
=   
dx dy dz 
0 0 0 1  xyz
2 .02423156789529607325... 


3 .02440121749914375973... 
.02461433065757607149... 
 ( k  3)

e1/ k


3
k!
k 0
k 1 k

4  3 (3)
log 2 x dx
  5
16
x  x3
1

(2 k  1) 2

4 
k  0 ( 4 k  1)

1 .02464056431481555547... 
.02475518879810900399... 
2 2 (1  log 2) 
61  6
72

.02481113982432774736... 
2
k 1
7 .02481473104072639316... 
.0249410205141828796...

.024991759249294471972... 
k
2

4
72
 64 log 2  12 log 2 2  22 (3) 
k 1
k
k 1

 k ( k  1)

2
3
Hk

(2 k  1) 2
1
( k  2 )( k  3)
Hk
k (k  1)(k  2)(k  3)
 5

 (k )
 5 
  k
log    
 6 6
k 2 6 k
43 log 2
log 

2 
 3 log 3 
 3 (1) 
4
12
2
(1) k  (k )  1

2k k (k  1)
k 2

.02500000000000000000
=
1
40
.0252003973997703426...

3
 ( 2 )   ( 3)   ( 4 )  
2


 
=

k
k 2
5
1

 k4
 ( 1)  ( k  4)  1
k 1
k 1
.02533029591058444286... 
.02540005080010160
=
1
42
1000
3937

.02553208190869141210... 


= meters/inch
 ( 3k )  1
8
k 1

k

 8k
k 2

1
3
1
8
4430
k
  ( 1) k k
2187
2
k 1
1
.025641025641025641
=
39

( 1) k 1 3k
3
3
1 .02571087174644665009...  2
arcsinh
 
7
2
 2k 
k 1
k 
k
2 .02560585276634659351... 
1 .02582798387385004740... 
2
8
csc 2



cot


2 4 2
2

k (2k  2)
= 
2k 1
k 1

 (k )
1 .02583714140153201337...   ( 6) ( 7 )   1 7 
k
k 1
.0258653637084026479...

.02597027551574003216... 
1 .02617215297703088887... 
 ( 2k
k 1
.026242459644248737219... 
J840
HW Thm. 290
6
2( 5)  ( 2 )(3)  ( 4 )  (3)  ( 2 )  4 
1


( 2  1) 
2 16

4 2 2


8
csc

1
 (8k  1)(8k  1) 
k 1

8


x
0
dx

1
8
161 G

   x 5 arccot x log x dx 
900 6
0
1  2 
128


  k64(2k )
24 

k 1
k
Hk
 (k  2)
k 1
1
.026227956526019053046... 
1

 1) 2
2
k 1
   ( x)  1dx

6
k  2 k log k
5

.02584363938422348153... 

1

2
4

GR 0.239.7
1

38
1
7
.02632650867185557837... 

2
2e 2
.0263157894736842105
=

 ( 1)

k
k 0
2k k 2
( k  1)!

3k k 2
3
241 .02644307825201289114...  12 e  

k 1 k !

 .02648051389327864275... 
.02649027604107518271... 
.02654267736969571405... 
=
1 .026697888169033169415... 
=
1 .02670520569941729272... 
.02671848900111377452... 
.02681802747491541739... 


2
1
log 2    k

3
k 1 2 (8k  24)

3 G
( 1) k 1 k

 
3
64
2
k 1 ( 2 k  1)

3 ( 3) 7
( 1) k


 
3
4
8
k  0 ( k  3)
1 1 1 2 2 2
x y z
0 0 0 1  xyz dx dy dz
 
 
log
csc
 
2
 2

1 

  log1  2 
 2k 
k 1

 ( 2k )
k 1
2k k

dx
1 x( x  1)4 
2

81 3
2
5
dx
 x4
Hk
k
k 1

x

4 ( 7 )   ( 2 ) (5)   ( 3) ( 4 ) 
 ( , ) 
5 3

6
13 (3)
1  ( 2)  7 
 2 
     ( 2 )    

18
216 
 3 
6


 
=

 
=





10 3  (3)
1
 4i Li3  1  i 3   2 3  i Li3  1  i 3   


 2 
 2 
2
81 3
3( 3  i ) 





log 2 x dx
1 x5  x2 

2

.026883740540405483253... 
1
12
3
log 
5
2
(1) k 1

k 1  2k  k
  6 (2k  1)
k 


1
1 ( 3)  1 
6 .02696069807178076242... 

    6
4
81
 3
k  0 (3k  1)
1
.027027027027207207027 =
37
1
log3 x dx
0 x3  1

.02715493933253428647... 
.02737781481649722160... 


 (k )  1
k 2
2
(2 k )!



4
64 768
3 5
.02737844362978441947... 


2 32
.02742569312329810612... 
1 .0275046341122482764...


( 1) k

2 
k  0 ( 2 k  5)

8
.02707670528833012617...  G 

9


log x dx
x6  x4
1

1
   1  2 k  cosh

k 2

k ( k  1)
 2(2 k  1)

k 1
4
1 

k

J349
 /4
 sin
4
x tan 2 x dx
0
  
1
 Li2 (1  e) 
4

B2 k
 (2k  1)!
k 0

.02753234031735682758... 
x log x
3 log 2 2
 log 2   
dx
2
2
0 cosh x
.02755619219153047054... 

  (5)

 (5)

p prime
log p
1  log k



p5  1
 (5) k 1 k 5
(k )
6
k 1 k


 
3 .0275963897427775429...
=

.02768166089965397924... 

cosh  
8

289

.02772025198050593623... 

k 2
1 .02772258593685856788... 
 (k )
k
6


0
e

 e
2



k
 (2 k )! 
GR 1.411.4
k 0
x sin x
dx
e4 x



 1
1
  Li  k   k 
k 1
6

 (1) k 1
3 3
(1) k 1 


  
3
3
64 2 k 1  (4k  3) (4k  1) 
(1) k / 2 

3
k  0 ( 2k  1)

1
1
.02777777777777777777 =
 
36
k  0 ( 2 k  1)( 2 k  2 )( 2 k  4 )( 2 k  5)

k
=

k  0 ( k  1)( k  2 )( k  3)( k  4 )



( 1) k
(k )
(k )
 
 
.027855841799040462140...   k

k 2 4  (k )
k 1 4 k  1
k  1 4 k ( 4 k  1)

15 (5)
( 1) k
.02788022955309069406...  1 
 
5
16
k 0 ( k  2)
J326, J340

=
Prud. 5.1.4.3
K ex. 107c

1 .02797279921331664752...
2 .02797279921331664752...
19 .02797279921331664752...
1 .0281424933935921577...

.02817320866780299106...
k2
 7 e  18  
k  0 k !( k  3)

k4
 7 e  17  
k  0 ( k  2 )!
 7e

1 

sinh k

 (2k )  1
csc  e   
 1  log  sin
2 
k
e
k 1



3 (3)
log 2 x dx
x 2 dx
  5
  4x

128
x x
e 1
1
0
1 .02819907222448865115... 
.02835017583077031521... 

 
=
.02839399521893587901... 
.02848223531423071362... 
.0285714285714285714
=
.02857378050946295008... 
.02874500302121581899... 

  1

1 .02876865332981955131... 


c  2 k 1
.028775355933941373288... 
.02883143502699708281... 
2
 
.0290373315797836370...
=
k
2






 1
c

  Li  k
c 1 k  2
k 1 3
(1)
2
k
 (k  1)!2
k 1
3 (3)
2
 log 2 

2
4
k

log 3 x
1 ( x  1) 4 dx
1
2
3  log x dx 

 log1  x
0

 (ck )  1
13

4 e
1
18 .02883964878196093788... 

1
7

 8 log 2  2 2 log 1  2  
  cot
16 
8



 (k )
1
 

k
2
k 2 8
k 1 64k  8k

1  2 cos1
1  12 (cos1)
cos k

  k 
GR 1.447.1
5  cos1
4 cos1  5
k 1 2
4

3 4
( 1) k 1 k

 
2 e
k 1 ( k  3)!
1
35

log k
  (5)   5 
k 1 k
1

2
19
log 2 log   (3)





  x 2 log 2 x arctan x dx
54 108 216
27
2
16
0
12 
2
1
 7
 2
 2 1 
 7
 4 log 2   (1)     (1)      ( 2 )     ( 2 )   
 6 
 3
 3   72 
 6
6
3
1  (2)  9 
(2)  2  

        
 10 
 5 
1000 

log 2 x dx
1 x5  1 
.029076134702187599206... 
.0290888208665721596...

3 (3)

4 3

e 
0


108

dx
2 x1 / 3
 (x
2
0
1
dx
 9) 2

sinh  3
3 

  1 

( k  2) 2 
7 3
k 1 
1
.029280631484544157522... 
19 log 2  3  6log   6 (1)  2 
12
3 .02916182001228824928... 

 
=


k 2


 4
k 2
.02941176470588235294... 
.029524682084021688...

 
k (k  1)


1
1


48 16e 2

log k
  5
k 2 k  1
9 .02932234585424892342... 
k
 1  4k  (2k  1) log1  2k  
.02929178853562162658... 
.029292727281254639...
k 2
1 

1
 (k )  1

2
k

( 1) 2
k
k
 ( k  4)! 
k 0
( ( k )  1) 2  1
1
34

c3 


1 4
  12 2 (2)  8 (3)  3 2 (2)  6 (4) 
24
Patterson ex. A4.2
7 (3) 56
1

 2
3
4
27
k  2 ( 2 k  1)

.0295255064552159253...


=

1
1
=

0
.02956409407168559657... 
=
1 .02958415460387793411... 
.029648609903896948156... 
log 2 x
dx
x6  x4
x 4 log 2 x
dx
1  x2
GR 4.261.13
1
1  (1)  5 
 1 

     (1)    
 6
 6 
4 36 
1  (1)  5 
(1)  4  
        
 6
 3 
36 
( 1) k 1

2
k  1 ( 3k  2)

log x dx
x6  x3
1

 1
 3 3 1
HypPFQ  ,1,1,  , ,  
 2   2 2  8 
1 9 (3)


6
8 2
 ( 2k )



1
k 0
  2k (2k  1) 2
k 
  2k 
 4 (2k  1)(2k  3)
k 1
k
1 .02967959373171800359... 

 .029700538379251529...

.029749605548352127353... 
1 .0297616606527557773...

 3/ 2
log x dx
1  3
    1     
4
16  4   4 
0 1 x
 2  9
24

( 2 k  1)!!

2
9


3 8
 (2 k )! 4
k 2

 
 log 3 
 
3 3
3 3

log x dx
1  x3
 2 39


4
16
3
k

9 log 2

4
 
1
=
1

 (k )   (k  1)
4k
k 2


x dx
3
0
e3x  1

GR 3.456.1

GR 4.244.2
0
.0299011002723396547...


1
  k ( k  1)( k  2)
k 1
2

J241, K ex. 111

3 log 2 log 2 2 5
Hk


  k
2
2
4
k 1 2 ( k  1)( k  2 )( k  3)

2
3
H k2
.0300344524524566209... 
log   ( 1) k k k
2
27
2
k 1

 (k )
 6 
.0300690429769907224...  log    
  ( 1) k k 
 5 5
5 k
k 2
.0299472777990186765...



 
=
k 1
1 .0300784692787049755...

 1
1 

  5k  log1  5k   


e 

3
e   , 0,1 
erf 1  1 
2

2

.03019091763558444752... 

 
=
=
.03027956707060529314... 
(2k )!!
k 1
e 
k!4
erf 1  

2
k  0 ( 2k  1)!
1 1
 (3)      1  i    1  i  
2 2
1
 (3)     (2  i )   (2  i) 
2


1

(1) k 1  (2k  3)  1


5
3
k 2 k  k
k 1

2 .0300784692787049755...

 (2k  1)!!k!(2k  1) 
3
1024
k


x 2 dx
0 e4 x  e 4 x 



16G  2 2
1 .030296955002304903887... 


log 2  3 
3
9
3
1/ 2
arcsin 2 x dx

x2
1 / 2
.03030303030303030303
=
1 .03034552421621083244... 
1 .03034552421621083244... 
.03044845705839327078... 

1
33
6 ( 3)
7

e
e
 ( 3)
    ( 2 )  c2 
42
j5 
1
1
1
1
cos 
sin 
.03060200404263650785... 
64
4 16
4
1 .03055941076...
2 .03068839422549073862... 
2
2
 1  2( 3) 

=
 ( 1)
k
Berndt 9.27
J314
( 1)

k 1
.03087121264167682182... 
.03095238095238095238
=
.03105385374063061952... 
k

5k  5k  2
3
k  2 k ( k  1)


( k 2  1)( ( k )  1)
k 2
.030690333275592300473... 
k
 (2 k )! 16
k 1
2
2
 1
1

 log   2   




4
3
x3

 ( 1)
k 2
k
 (k )  1
 kk
   2x 2x
dx
240 8
 1
0 e e

13
1
 
420
k  0 ( 2 k  1)( 2 k  5)( 2 k  9 )
3


( 1) k 1
1
 
3
32
k 1 ( 2 k  1)

33  2 9( 3)
( 1) k


 
3
48 48
24
k 1 k ( k  2 )

15 (5) 45 (3)
( 1) k 1
.031125140377033351778... 

 140 log 2  126   5
5
8
2
k  1 k ( k  1)

log 2
1 
 (k )  1

  ( 1)  
.03117884541829663158... 
4
6
k  2 k ( k  1)( k  2)
2 k 1


2 (2 k )
1
2
5 .03122263757840562919...  
  sinh
k
k 1 ( 2 k  1)!
k 1 k
2

( 1) k 1 k

.03122842796811705531... 
 2 log 2  3  
2
6
k 1 ( k  1)( k  2 )

1 
(1) k 1
.03124726104771826202... 

 csch 4   2
32 8
k  1 k  16


 (5k )
1
.03125002980232238769...   k   k
5
k 1 2
k 1 32  1
.03111190295912383842... 
.03125655699572841608... 
1 .03137872644997944832... 
.03138298351276753177... 
1 .03141309987957317616... 
.03145390200081834392... 
2
 1
 2 log 2 2  8 Li3  
 2
3

1

5
k 1 ( k !)

35 2
1
4
126 

  5
5
3
9
k 1 k ( k  1)

1
1
cosh  
k
4
k  0 ( 2 k )!16
1   5
1  (1)  5
 9
 9
   2,     2,   
     (1)   
 8
 8
 8
64   8 
64 
2
( 1) k 1

2
k 1 ( 4 k  1)

log x dx
=  6
x  x2
1

=
.031667884637975851999... 
cos1sin 1  ci2 
.03202517057518619422... 
1  log
 
=
 ( 1)
k 1


cos x sin x
dx
x2
1


2
 coth  
sinh  2



k


   k
k 2
2
1 
1

 log 1  2   

k 
1
k 1
 (2 k )  1 
k
2k
1
 .032049101862383653901... 
  r (k ) 1  14k  76k  168k   
3

8
k 0
1
1

 


Borwein-Devlin. p. 36
where r (k )    k  
2   (k  1)


32
7
2
3


.032067910200377349495... 
.032080636429384244827... 
=
.032092269954397683...

.03215000000000000000
=
.03224796396463925040... 
 2 5 log 2
1



6
64
12

log x
 ( x  1) ( x  1) dx 
4
1

3
1 
1 
 64 2 G  2976 (5)    ( 3)     ( 3)     

4096 
 4   
4


 /4
0
x 3 log tan x dx
527
2 log 2


7350
35

1
log 2 x
dx
 
32
x5
1
12 log
3 29


2
6
( 1) k 1

k 1 k ( k  2 )( k  7 )

( 1) k 1 k

k
k  0 2 ( 2 k  6)

.03225153443319948918... 
.032258064516129
=
.032309028991669881698... 
.03233839744888501383... 
.032355341393960463350... 

.032411999117656606947... 
.03241526628556134792... 
1
3
1
31
MHS (3,2,1)  3 2 (3) 

2

8
2  
 
=

e
x
x log x dx 
0
42 2 

8



0
x 2 (1  x)
dx
x8  1
 /4
1

8 4 2 log 2  9 (3)  24 G    x 2 log cos x dx 
0
6144
(5)  3 ( 3)  2( 4 )  5 ( 2)  15 


 2 log 2  
203
 (6) 
48
k
k 1
2
1
( k  1) 5

5 2
1

 
2
2
2
4
k 1 k ( k  1) ( k  2 )

 2k 
 log 2 G
1
1 .03263195574407147268... 

   k
Berndt 9.6.13
2 
4 2
2
k  0  k  8 ( 2 k  1)


1
1
242 (3)
 G5  1  

1 .0326605627353725192... 

J309

5
5
243
k 1 (3k  1)
k 1 (3k  1)

1
1 .03268544015795488313...   k  3 
k 1 k
.03259889972766034529... 

 .03269207045110531343... 
.03272492347489367957... 
3 5

4
6

96


x
0
6
dx

 64

19 log 2
( 1) k 1 Hk

 
72
3
k 1 ( k  2 )( k  4 )

2

log x dx
  6
.03289868133696452873... 

x x
300
1
.03283982870224045242... 
.03302166401177456807... 
.0331678473115512076...

J1  4

4
36 log

4k
( 1)

k !( k  1)!
k 0
5
8 
4
k
( 1) k 1 k

k
k  0 4 ( k  1)( k  3)

7( 3)
4


16
192

2k
2 .03326096482301094329...  
6
k 1 k ! k
1 .03323891093441852714... 
.033290128624160141531... 

3

 
=
k
 (2 k  1)
k 1
4
 3  i 3
 3  i 3
3 2 1
  (1  i 3 ) 
 

  (1  i 3 ) 
2 6 6
 2 
 2  



 ( 1)  ( 3k  2)  1
k 1
k 1


k
k 2
5
1
 k2
1
 B2  B4
30

1
1 .03348486773424213825...  
4
k 1 k ! k
.0333333
=
1 .03354255600999400585... 
.03365098983432689468... 
=
3
30
1 (1)  1 13 (3) 2 3 3
  

 3
27
81
729

1  (1)  1
k
( 2 )  1 


6


 
 


3
 3
 3 
162 
k  1 ( 3k  1)
.033749911073187769232... 

5

 3 log 2 
4
3
.03375804113767116028... 
5   ( 2 )   ( 3)   ( 4 )   (5) 
log(1  x 6 )
0 (1  x)2 dx 
1
.03388563588508097179... 
1 .033885641474380667...

.03397268964697463845... 
.03398554396910802706... 
.03399571980756843415... 
.03401821139589475518... 
1
 k5
k 2

( k ) 0 ( k )

2k  1
k 1

pq ( k )

2k
k 1
3  2 log 2 log 2 2
 1



 Li3  
 2
2 12
2
2


 (2 k  1)  1
1
 1
     sinh 

 k
(2 k  1)!
k
k 1
k 2

1
( 1) k
J 4 ( 2) 
F
,
5
,
1




24 0 1
k  0 k !( k  4)!
k
1
 k ( k  1)
k 1

=

5
6
1  (1)  5 
(1)  11  
        
 12  
 12 
144 

log x dx
x6  1
1

LY 6.117
.03407359027997265471... 

log 2 5
1

 k

2
16 k 1 2 (8k  16)
 /4
 sin
4
x tan x dx
0

 2k 
 1
1
1

HypPFQ  , 1, 1, 2,2,      k
2
4

k  0  k  16 ( k  1)
 2

1
3 log 3
.03421279412205515593... 

 2 log 2  3  
3
2
k 1 36 k  k

1
1
 1

1 .034274685075807177321... 
 e    erf 1  erfi1   sinh 2  dx
x 

2
e
1
1 .03421011232779908843... 
1 .03433631351651708203... 


3 
( k !) 2 4 k

log 2  3  
2  1  k 1  
 4 
4
k  0 ( 2 k )!( 2 k  1)


J376
Berndt Ch. 9


dx
6
1 .03437605526679648295... 
csc
  7
x 1
7
7
0
1
.03448275862068965517... 
29

1 1
( 1) k 1
.03454610783810898826... 
 
e 3
k 1 ( k  3)!
2 .034580859539757236081...  12 log 2  2  

1
 ( k  1)( k 

)
2

3 (3)
( 1) k
3 
.034629994934526885352... 

 2 log 2 
  4
3
4
2 12
k 2 k  k

1  ( 2)  7 
( 1) k 1
( 2 )  3 


.03467106568688200291... 


 
 


3
 8 
 8
1024 
k  1 ( 4 k  1)
k 0
1
1 .03468944145535238591... 

 

 


(sinh  ) cosh   cos  3

2 2
sinh 
 cosh   1 1  cos 3  1  cosh   1  cos 3 
=
4 3

1

=   1  6  
Berndt 4.15.1

k 
k 1
2 .03474083500942906359... 


 1  e 1/ k
1 
1 .03476368168463671401...   
 2 1/ k  
k
k e 
k 2 


   (2 k  1) 
 Re

ik
 k 1

1/ k

1 e
1 
2 .03476368168463671401...   
 2 1/ k  
k
k e 
k 1 
 
4
sin x
 arctan x dx 
0
3



k2
( 1)
 ( k )  1 

k!
k 2
k
=

 ( 1)
k 2
k
k
 ( k )  1 
( k  1)!

.03492413042327437913... 
Ai ( 2)

1
H ( 4) k
1 
1 .03495812334779877266...  2 Li4     k

 k 
4
 2  k  0 2 (k  1) k 1 2
.035034887349803758539... 

 
=
.035083836969886080624... 
.0352082979998841309...

 
1 .0352761804100830494...

=

.03532486246595754316... 
.03536776513153229684... 
.03539816339744830962... 
.03544092566737307688... 
.03561265957073754087... 
.035683076611937861335... 
.03571428571428571428
=
.035755017483924257133... 
.03577708763999663514... 
1 1   1  i
 1  i
  1  i
  1  i
  
 
 
 
 
 2
 2 
 2 
2 4  2 


1
k 1

( 1)  ( 4 k  1)  1   5

k 1
k 2 k  k

1 
1
 (k )  1

 log   2    



 kk
k 2
 ( 4) 3 ( 6)
 2 ( 3)
( 2 )   ( 3) 

  ( 2) ( 3) 
 2 (5) 
4
4
2

Hk

5
k 1 k ( k  1)
5
2 3  1  csc 
AS 4.3.46, CGF D1
12

11   2
( 1) k
  2
3
32
k  2 ( k  1)
1
9

3
( 1) k 1

J244, K ex. 109c
 
4
k 1 ( 2 k )( 2 k  1)( 2 k  2 )
1 (2)  4 

  
 5
125
 (5)  1
 (5)

1    i 
(1) k  (2k  1)

   i 
 

  
   
 2 k 1
 2    
  
k 1
1
28

1
( k )
log  (5k )



5
k
p prime p
k 1



2

25 5

( 1) k ( 2 k )! k 2

( k !) 2
k 1

( 1) k ( 2 k )! k 3

( k !) 2
k 1



1 .03586852496020436212... 
 k  (4k  2)   (4k  1)   (4k )   (4k  1)  4
k 1

1
.03596284319022298703...   5

k 2 k  1
.03619475030276797061... 
H ( 3) k (k  2)

k
k 1 2 k ( k  1)


2
32


4
384


 ( 1)  (5k )  1
k 1
k 1
7 ( 3)

16

k2


4
k 1 ( 2 k  1)
.03619529870877200907... 
.03635239099919388379... 
.036382191549123483046... 

2    7 log 2

8

1  dx
2
 ( x  1) 3
 log1  x
1

( 1) k

2 k 1
( 2 k  1)
k 1 2
1
1
 arctan 
2
2

  sin(sinx 2 xx)cos x dx 


1
1
 
 log 1  2 
2 4 4 2 2

3/ 2
0
Prud. 2.5.29.27

.036441086350966020557... 
k
k 2
6
k

1

 ( 1)  (6k  1)  1
k 1
k 1
.03646431135879092524... 
1
6  1  H ( n) k 
log       k 
5
 5 k 1 6 
n 1 5
.03648997397857652056... 
 3
 1
2    2 log 2          
 2
 2

 (k )  1
1
1 
1

6k 
2k
k 2
k 2


 (k )
1 1
 
 4 
  k     log 1     
.03661654541953101666...  log    
 5k  5k 
 5 5
k 2 5 k
k 1 

.036557856669845028621... 
 2 (2k  1)
1 .03662536367637941437... 
1 (1)  1 
   
 6
36
k



 

2k arctanh
1

2 
k  1 ( 6k  5)
1
log x dx
x6  1
0


2
( 1) k 2 2 k 1
.03663127777746836059... 
 
e4
k!
k 0
2


k
.036683562016736660896...   7
  (1) k 1  (7 k  2)  1
k

1
k 2
k 1
3

3 1

( 1) k 1
.03676084783888546123... 


 
2
3
128 32 2
k 1 ( 4 k  1)

.03681553890925538951... 
3
dx
 2
256 0 ( x  4) 3

 ( 2k )
1 7 (3)



k
2
4
4
k 1 4 ( 2k  1)( 2 k  2)
1

.03690039313500120008... 
  2 log 2 
5 2

1  p 1  p 2  p 3
1 .03692775514336992633...   (5)   ( 4) 

1
 p  2  p 3  p  4
p prime 1  p
.03686080059124710454... 

(1) k 1
5  (1) k 1 k 1 1
 
 
2 k 1 3  2k  j 1 j 2
k  2 5  2k 
k  
k  
k 
k 


 

2
Borwein-Devlin, p. 42

 

 (4)

p prime
.03698525801019933286... 

.037037037037037037037 =
1
27
1  p 1  p 2  p 3
1  p 1  p  2  p 3  p  4
6

.03705094729389941980... 
.037087842300593465162... 
.03717576492828151482... 
.03717742536040556757... 
2 .03718327157626029784... 
.03722251190098927492... 
.037286142336031613937... 
.03729649324336985945... 
1 .03731472072754809588... 


1
 (8k  3)  1 



3
5
k 2 k  k
k 1


 (3k )
1
  k

3k
k 1 3
k 1 27  1


1
 (7k  2)  1



2
5
k 2 k  k
k 1


 ( 6k  1)  1
k log(1  k 6 ) 




k
k 1
k 2
32  3 
 
5  1 2 
1
1
3 

 ( 2)    

3
8192
 16  k  0 (16k  3)

1
k 2  2k  2
csch  sin  3   2
3
k 1 k  2 k  2


3 7
(1) k
1
9
  k
log    k 2
2 8
4
k  2 2 ( k  1) k
k 1 3 ( 4k  8)

coth 2  csch 2  tanh 1 
e 2  e 2

e2  e 2
1  16 k
B2 k

2 k  0 ( 2 k )!

1 3 2
1
.03736229369893631474... 

 

2
3
2
64
k 1 ( 4 k  1)
 
=
.03741043573731790237...  1  2 ( 4 )   ( 3) 

 

 
1728

k
4
7 (3)
1
1
 
 ( 2)   
432
3456
 4



1
 (12k  9)
k 1
 3  i 3 
1 1   3  i 3 

  
  
 2  
3 4 6   2 




1
=  5
   (6k  1)  1 
1
k 2 k  k
k 1

1
1  1
(1) k 1  (3k  1)  1   4
=

2 k 1
2 k 2 k  k
.03742970205727879551... 

3
J373, K ex. 109d
 ( k  2)
k 1
.03742122104674100704... 
AS 4.5.67
3


 (5k )  1
 e

k 5
1
k!
k 2

 ( 5k )  1
.03743548339675301485...  
   log(1  k 5 ) 
k
k 1
k 2
.03743018074474338727... 


k 1

3 log 2 19 
.03744053288131686313... 



4
20 8
.03756427822373732142... 

 .03778748675487953855... 
27 .03779975589821437814... 
.03780666068310387820... 
.037823888735468111...

.037837038489586332884... 


=

 
=
 (3)
32



1


k  2 4 k ( 4 k  1)

 ( 1)
k 2
k 1
( k )  1
4k


log 2 x
x2
dx

1 x 5  x 0 e 4 x  1dx


1
 16

48 3
k 1  2k 
 
k 

 (x
2
0
dx
 3)3
 6
1
1
1 
 ( 2)    
3
1458
 3  k 1 (9k  6)

16 29
( 1) k

 
k
3
e
k  0 ( k  4 )! 2

3 ( 3)
9
( 1) k

 2 log 2 
  5
3
4
4
k 2 k  k

(1) k 1


3
k 1 k ( k  1) ( k  2)

1


k
2
k 1 2 3k  9 k  6



1 2 log 2
( 1) k

 

3
2
3
k  1 27 k  3k


1
 (5k )  1



5
k 2 k  1
k 1


 ( 4 k  1)  1
log(1  k 4 )




k
k
k 1
k 2



1
8

 cos  2   1 
2 
7
(2k  1) 
k 1 

.037901879626703127055... 
.0379539032344025358...

.03797486594036412107... 
.03803647743448792694... 
1
1
 3 
 ( 2)    
3
1024
 8  k 1 (8k  5)
.03804894384475990627... 

.03810537240924724234... 
1
log 2




2
2
4
.038106810383263487993... 
[Ramanujan] Berndt Ch. 2
G
11


8 144

 xe
x
log x sin x dx
0
(1) k

2
2
k  0 ( 2 k  1) ( 2 k  5)

Prud. 5.1.21.42
1 . 03810687225724331102... 

  Hk
1
 (10)   (5) 2 
2

1 .0381450173309993166...

=
.0381763098076249602...

.03835660243000683632... 
( 5)
k 1

k
5



1
 1

   9  5  i 2 5  5      9  5  i 2 5  5  




4
 4

1

  
  9  5  2i 5  5     1  9  5  2i 5  5  
 
4
2   4 
2 

  
 
5

k

5
k 2 k  1

1
80  4  2  48 log 2  6 ( 3)   3
3
k 1 k ( 2 k  1)


1  log 2
1
1
 


k
2
2
8
k 1 32 k  16 k
k 1 2 (8k  8k )
1
=

log(1  x)
 ( 2 x  2)
2
dx
GR 4.201.14
0
(1) k 1

k  0 12 k  11

1 .03837874511212489202... 

.0384615384615384615
=
.0385018768656984607...

=
31 .03852821473301966466... 
.0386078324507664303...

1
 cos 5 x e  x dx
26 0
sin 2
 cos 2  sin 1 cos1
cos 2 
2

(1) k 1 4k k 2
  J1 / 2 (2)  J 3 / 2 (2)   
(2k )!
k 1
3
3
 

1
1
x dx




 
2
2
2  0 (1  x )(e 2 x  1)
1
=
 4
0
Andrews p. 87
log x
dx
2
 log x x  1
GR 4.282.1
2
.038622955050564662843... 
1  (1)  1 

 2   log 2
     (1)    


108 
9
9 3
6
 3 
.03864407768699789287... 
31  3 2
log x
 4
dx
36
(

1
)
x
x
1
(1) k 1 k

2
k 1 (3k  1)



1 .03866317920497040846... 

8

3
48
 /2

x
2
sin 2 x dx
0
7

21 ( 3)
28 .03871670431402642487... 

 log 2 

8
16
2
2
4
64 k 3  20k 2  8 k  1


2 k ( 2 k  1) 4
k 2

k 3 ( k )

2k
k 2


 
=
(1) k 1 k

3
k 1 ( k  2)

.03876335736944358027... 
2   (2)  3 (3)  2

1

 
=
.038768179602916798941... 
.03879365883434284452... 
.03886699365871711031... 
.03895554778757944443... 
.038978174603174603
=
.0390514150076132319...

.03906700723799508106... 

 
=
x 2 log 2 x dx
log 2 x dx

0 ( x  1)2 1 x 2 ( x  1)2
 (3)
3
1
8
  2 log
2
2  1  

1
0
2
1
dx

2
2
 16 log x x  1

1 1
5
( 1) k 1
 log  1   2 k 1
k ( 2 k  1)
2 2
4
k 1 2

7 ( 3)
1
 
2
216
k 1 ( 6 k  3)

3929
1
 
100800
k 1 k ( k  5)( k  8)

3
 (3k  2)  1  1
  2 1  ek

k!
k 1
k 2 k
1
3  4  2 ( i )  2 (i ) 
8

5 
1
1

   (2  i )   (2  i )    5
8 2 4
k 2 k  k
2 arctan




 
=
  (4 k  1)  1
k 1


 ( 3k  2)  1
log(1  k 3 )

k
k2
k 1
k 2

 2 21
k
.03915058967111741409...  1 
  ( 3)  
3
16 16
k 1 ( 2 k  3)

1

2
( 1) k
.039155126213126751348... 


 
2
2
32 48 3 216
k  0 ( 3k  1) ( 3k  5)

25 log 5
11
( 1) k 1
 25 log 2 
  k
.03929439142762194708... 

2
4
k 1 4 k ( k  1)( k  2 )

1
 
=  k
k 1 5 ( 2k  4)
.0391481803713593579...

.03939972493191508632... 
21  2 2

32

 

 (k
k 2
2
1
 1)3
GR 4.282.9
.03945177035019706261... 
.039459771698800348287... 

12   2
(1) k 1
 2

2
54
k 1 (3k  3)
2 log  
  k
1 .03967414409134496043... 
log(1  x 3 ) log x dx
0
k 1
 k  1 (2k )  1
k 2
ei  e i
30

2

5
( 2)


1
 coth    2csch 2  1
2

=
2
2
2 cos1 
1 .03952133779748813524... 
3 .03963550927013314332... 
x
k 2

2
 2 log(1  k 2 )  
2
1 k
k 2 


=
9

4

1

 (1)k 1 2k  (2k  1  

2

2
k 1  k 2

1
3 log 2
1
1 .03972077083991796413... 

 3 arctanh   k
3
2
k  0 9 ( 2 k  1)

1
 
= 1  2
[Ramanujan] Berndt Ch. 2
3
2 
k  1 64 k  4 k

1 ( 2)  3 
1
.03985132614857383264...  
    
2
250
5
k  0 (5k  3)
k 1

.03986813369645287294... 

 
=
2 .03986813369645287294... 


2
k 2

2
(1) k 1 k  (k  3)  1 

3
k 1

4
1

2
3 k 1 k (k  1) 2 (k  2) 2
2

13

4
5


3
4
(1  x 2 ) log x
0 x  1 dx 
1
.04000000000000000000
=
1
25
.0400198661225572484...

 (3,4)   (3) 

251
1 log 2 x
  5
dx
216
2 1 x  x4
1 1 1

 
=
x3 y 3 z 3
0 0 0 1  xyz dx dy dz

.04024715402277316314... 

 ( 2 k  1)
k 1
1 .04034765040881319401... 
4 .04050965431073002425... 
.040536897271519737829... 
.040634251634609178526... 

 
=
.04063425765901664222... 
1 .04077007007755756663... 
.04082699211444566311... 
=
.04084802658776967803... 
.04101994916929233268... 
1 .04108686858170726181... 
.04113780498294783562... 
.04132352701659985...

.04142383216636282681... 

3 10
2k

 (4)
7

 (3) 2
Hk
3 (6)

 MHS (5,1)  

5
4
2
k 1 ( k  1)


13 
1
 ( 2k )  1
 



2
30 8 k  2 16k  1
16k
k 1
1  (1) k

2 k  0 2k  7

1 1
1
( 1) k k
sin  cos  
k
2 2
2
k 1 ( 2 k  1)! 4

23 sin 1  5 cos1
k4
  ( 1) k
( 2 k  1)!
16
k 1

Titchmarsh 1.2.14
 (3)  1 2   f 2k(2k ) 
k 2
 
1


3
k  2 k  2 ( jk )

 3 log 3 log 2
( 1) k ( k )
1 
1
1


  2
 
2
4
3
6 k 1 6 k  k
6k
k 2

1
1 383
( 1) k 1
168 cos  92 sin 
 
k
2
2
2
k 1 ( 2 k )! 4 ( k  2 )
 Li1 e 2i  Li1 e 2i 



x
4
dx

 81
54 2
0

B2k

k
k 1 ( 2 k )! 2
1 1
e 

 tanh 

2 2
2 1  e 

 (1)
k 1
k 1  k
e
J944
.04142682200263780962... 
1 .04145958644193544755... 
.0414685293809035563...


7 (3)  3
1 ( 2)  3
1
    

 
3
 4
16
64
128
k  1 ( 4 k  1)
 2 log3 3
1
 1

 2 Li2    Li2   
6
2
 3
 3
 3 
1 2 e
1

cos
  
3e
3
3
 2
2



( 1) k 1

k  1 ( 3k  1)!

.041518439084820200289... 
( 1) k
1
  tanh  coth    4
2
12 4 
2
2
k 2 k  k
.04156008886672620565... 
26 (3)
4 3
1
 4

 2    (2)   
 3
27
27
81 3


1
log 2 x
dx
x5  x 2
1
26 (3) 4
1
1

   ( 2 )    2
3
27
27
81 3
 3
k 1 (3k  2)
3
2 .04156008886672620565... 


1
=
log 2 x
0 1  x3 dx

=
.04156927549039744949... 
=
.04160256000655360000... 
.04161706975489941139... 
.04164186716699298379... 
2 .04166194600313495569... 
.04166666666666666666
=
=
x 2 dx
0 e x  e  2 x
i   3  i 3  1
i   3  i 3
1

  
  ( 2)   
 
 
 6 2 3  2   6 2 3  2 
3

1

5
2
k 2 k  k


 ( 2k )
1


k


2k
2
1
k 1 5
k 1 5

1 
1
1
1
1 
( 1) k 1 k
sin
 cos
sinh
 cosh
  
4 2
2
2
2
2
k  1 ( 4 k )!
k 1

1
1
( 1)
cos
cosh
1  
GR 1.413.2
2
2
k 1 ( 4 k )!

k

k 1 k ! H k

1
1
C132
 
24
k  0 ( 3k  1)( 3k  4 )( 3k  7 )

k
Prud. 5.3.1.2
(1) k 1  k

e  (1) k
k 1
1

=
1 .04166666666666666666
=

k 13
e k  (1) k
k 1

25
1
 
k
24
k 1 k ( 2  2 )
 (1)k 1

1 .04166942239863686003... 
k 1
1 .04169147034169174794... 
.04169422398598624210... 
.04171115653476388938... 

 ( 2  1)
8

1
 (k
2
)!
1
1
cos1 e
cos1  cosh 1 
 


2
2
4 4e

 
( k )
1
  j

k!
k 2
k  2 j  2 ( k )!
1 .0418653550989098463...


.04188573804681767961... 
1
 (4 k )!
k 0
 5  2 2  log 3
1
2

arctan 2 
log

16
8
4
5 2 2


2
arctan 1  2  arctan 1  2

8
.04171627610448811878... 




x
4
2
sinh 1  sin 1
e
1
sin 1




8
16 16e
8

2
3
e
 

cos 
3 3 e
2 
3
1
9
log 
6
7

x
4
2

dx
 x4

k
 (4k )!
k 1
1
 (3k  1)! 
J804
k 0
dx
 x2
( 1) k
.04189489811963856197...   5

k  2 k log k
3 6
721 .04189518147832777266... 
4

1
1
cos k

.04201950582536896173...   log 2  2 cos1   log 2 sin   

GR 1.441.2

k
2
2
k 1

9
( 1) k 1 Hk
2
.0421357083215798130...  4 log 2  log 2 
 
4
k 1 ( k  2 )( k  3)

1
e 1
1

 2(e1 / 2  e 1 / 2 )  
1 .0421906109874947232...  2 sinh


k
2
e
k  0 ( 2k  1)!4

1 

=  1 
GR 1.431.2

4 2 k 2 
k 1 


1
 

 
.04221270263816624726...    4  cot 4  coth 4  
2 4 2
2
2

1
 
= 

4
k  2 2k  1

1 .04221270263816624726... 

k 1
=
(4 k )
2
k


 2k
k 1
1
4
1
1
 

 
 4  cot 4  coth 4  
2 4 2
2
2

 (4 k )  1
k 1
2k



1 .042226738809353184449... 

1 
1
 1


2


2


k
k2
k ( k 1)
2 
 2k
k 1  2  1
k 1 2
[Ramanujan] Berndt IV, p. 404
2 .04227160072224093168... 
16   2 2 
 ( 1)
( 1) k 1 




2
( k  1 / 4) 2 
k  1  ( k  1 / 4)
k 1


.0422873220524388879...

.04238710124041160909... 
.04252787061758480159... 
Bk 2 k k 2

k!
k 0

log 2

1

 
4
24
k 1 ( 4 k  3)( 4 k  1)( 4 k  1)( 4 k  3)

1
1 
1 1 

   1      1     2 
  
 2  


=
2
k 1
 (2 k  1)
 2 k 1
k 1

1
 k  k 
2
 1



8
12
4
.04257821907383586345... 

log 
125 125
3
.042704958078479727...

.04271767710944303099... 
6

2
4
 2 log 2 

 
7 ( 3)

4


1
 k (2 k  1)
k 1
3
k
 (2 k  1)! 4
k 1

x
2
k
dx

1
4
 
 1    2 (2)  3 (4) 
3 
10 
2
1
1
csc 
2
2
1 .042914821466744092886... 

e x dx
 1  e  2 x

Marsden Ex. 4.9
( 1) k 1 k 2
k
( k  2)
k 1
1
2 3
(1  x)3
3
1 .04310526030709426603...  3    5 log 2 
  log
dx
log
1  x6
2
2 3
0

1

log 3
1
.0431635415191573980... 




6 12 3
4
k 1 ( 3k  2 )( 3k  1) 3k ( 3k  1)
.04299728528618566720...  16 log

k 2 Hk
4k
k 1
k 1
arctan 2  log 3
 

4
4
4
2
.0428984325630382984...
 ( 1)
1
1
1
cosh  sinh 
2
2
2
.04282926766662436474... 
J242
 

5 .04316564336002865131... 


2

GR 0.238.3, J252
e 

.0432018813143121202...
3 58


2
9

k 2
 (k )  1
k
4



 1
1
  Li  k   k 
k 2
4
4
2 1

2 1
.04321361686294489601... 
1
2
.043213918263772254977... 
e   i 2i  (1)i  cos(2 log i )  i sin( 2 log i ) 

.0432139182642977983...

Mel1 4.10.7
(21/ 4  1) 1 / 4

211/ 4  3 / 4

.04321740560665400729... 
4
e 
 k2

e 
 ( 2 k  1) 2

J114
k 1

k 1









1
1
1



log
tanh



 ( 2 k 1)
( 2 k  1)
e
2
2
k 0 e


(k!)3
2 4 1 
1 .04331579784170500416...  HypPFQ 1,1,1,  , ,


 
 3 3  27  k  0 (3k  1)!

 1 i 3 
 1  i 3 
1
1 / 3
  1  4 / 3    1  4 / 3   
1 .04338143704975653187...        1  2

3 
2
2






 ( 3k  1)
1
 
= 



k
4
2
k 1
k 1 2 k  k
.04324084828357017786... 
arctanh



23 13
5 log 2 2
kH k
 log 2 
  k

4
2
2
k 1 2 ( k  1( k  2 )( k  3)

2
2
1
.043441426526436153273... 

arcsinh1   k
3
2
k 1 2 ( 2 k  1)( 2 k  3)


k 1
k 1
.04344269231733307978...   ( 1)  ( 4 k )   ( 4 k  1)    5
k 1
k 2 k  k
1
.0434782608695652173913 =
23


 (k )  1
  2 2

.04349162797695096933...  
 Li2     

k 2
k k
k 2 2 k
k 2 
log 3 log 2 
1
 .043604011980378986376... 



3
4
3
k 1 27 ( 2k  1)  3( 2k  1)
.04341079156485192712... 
[Ramanujan] Berndt Ch. 2
1
1 .043734674009950757246... 
arctan(e  1) 
e
x
0
1 .04377882484348362176... 
( 4)

 (5)

( k )
k 1
k5

1
 2 e x  2

3 log 2 
1
.04382797038790149392... 
 

4
8 12
Titchmarsh 1.2.13

1


k  2 4 k ( 4 k  1)

( k )  1
k 2
4k


1 .04386818141942574959... 
.04390932788176551020... 

( k !) 2
16
1
arcsin  
k
4
15
k  0 ( 2 k  1)! 4
 /4
23
8
   sin 3 x tan 2 x dx
6 2 3
0
 ( 1) k 1
( 1) k 1 
7  216   

3 
( k  1 / 6) 3 
k  1  ( k  1 / 6)

1 .04393676209874122833... 
3
(1) k 1

k  1 ( k  1)! ( 2 k )

.04398180468826652940... 
Titchmarsh 14.32.1

1
( 1) k
 2  csch   2
2
k 0 k  1
 k

e
e 
1 .04416123612010169174...  e
 
k 0 k !

1
( 1) k 1 k 2  2 k 
.04419417382415922028... 
 
 
4k
k
16 2
k 1
1 .04405810996426632590... 
.044205282387762443861... 
2
.04427782599695232811...  2  2 log 2  3 
 4 log 2 
3

1 


4 48
2
.04438324164397169544... 


( 1)

k 1
 ( 2 k  2)
k 1
2

x
1
5


14
 log 2   (2)   (3)   (4) 
3
1
dx
 5


5
4
4
k 2 k  k
2 x  x

 2k 
  k  16
k 1
dx
 x3

9 .04441336393990804074... 
  ( k )  1 ( k )  1
k 2
2

2
arccosh x
.044444444444444444444 =
 
(1  x ) 4
45
1

 ( 3)
1
.0445206260429479365... 
 
3
27
k 1 ( 3k )

3 1
dx
.0445243112740431161... 
   2
( x  1) 3
32 4
1

9 log 3 5
1

  k
2
4
k 1 4 k ( k  1)( k  2 )


 1 1
1
5
1
.04462871026284195115... 
log    Lik       k 1
5
4
k
 5 5
k 1 
k 1 5
k 1 2

3
arcsinh 1
( 1) k  2 k 
.04482164619006956534... 

  k
 
4
8 2
k  1 4 ( 2 k  1)  k 


 ( k  1)
1 
1


log 1  
1 .04501440438616681983...  

k
 2k 
2 k
k 1
k 1 k
.04456932603301417348... 
9 log 2 
k
1
k (2 k  1)
.04507034144862798539...  1 
.04510079681832739103... 
1 .0451009147609597957...

1 .04516378011749278484... 
2 .0451774444795624753...

.04522874755778077232... 
.04527529761904761904
=
.04539547856776826518... 
1 .04540769073144974624... 
.04543145370663020628... 
.045452882429679849...
3


k
391 53e
144 

 
2e
2
k 1 ( 2 k  1)!( k  3)

1
1
 k
2 2 arctanh
2 2 k 0 8 (2k  1)
li( 2 )

7
H
8 log 2 
  kk 2
2
k 1 2

3 log 3
1
  2 k 1
1  log 2 
2
k (2k  1)
k 1 2

1217
1
 
26880
k 1 k ( k  4 )( k  8)
( 4 )   (5)


1
1

coth  3/ 2   2
2
2
k 0 ( k  )


2 log 2 5
(2k  4)!
1
 
  k

3
12 k  2 (2k )!
k 1 2 (6k  12)
 log1  p   p 
2

2
p prime
1
22
1
.04547284088339866736... 
7

7 ( 4 ) 3 ( 3)
( 1) k 1 k
.04549015212755020353... 


 
4
8
4
k 1 ( k  1)

17
dx
.0455584583201640717... 
 3 log 2   x x
e (e  1) 3
8
1
.045454545454545454545 =
.04569261296800628990... 

2


1
 ( 6k )
2
36
k 1

1
1
.0458716234174888797... 


log 2  1 
16 16 2
32 2
1
log x
dx
 
=  2
2
2
  16 log x x  1
0
.04596902017805633200... 
216

9



( 2)

 
=

18 3
1
 27 k
k 1

3
1


GR 4.282.10
3  3i  7  i 3 
3  3i  7  i 3 
log 3





6
18 3  6  18 3  6 

10
3 47
1
log 
  k
3
2 36
k 1 3 k ( k  1)( k  3)

 (k )
  5  
  ( 1) k k 
.04603207980357005369...  log     
  4 4
4 k
k 2
.04599480480499238437... 


 
=
 ( 1)
k
 (k )
1 .0460779964620999381...
 1   ( 2 ) 


k
4 k
1
.04607329257318598861...  I 3 ( 2 ) 

6
k 2
 
=


k 1

1 

 log 1   
 4k  
1
 k !( k  3)!

(2 k  2)
 1

  4k
Dingle 3.37
k 1
2

cot
2


 2k
4
k 1
1

 k2

2k

 log 2
1



 
8
2
k 1 ( 4 k  2 )( 4 k  1) 4 k
 
1
= 
2j
1
j 1 k 1 ( 4 k  1)
k 1
.0461254914187515001...
J251
J1120


 5
1

tan
  F2 k 1  (2k )  1 
2
2
5
k 1
 3i 3
 5i 3 
5 4 log 2  

.046413566780796640461... 

  cot
 cot

6
3
6
4
4


1
2 .04646050781571420028...   k

1
k 0 3  3
1 .046250624110635744578...

. 04647598131121680679... 
3  2G

8



( 1) k


4
k 2 k  k

2
 2k 
k
 


k
k  0  k  16 ( 2k  1)( 2k  3)



1  3
1

 
2
12
k 1 ( 6 k  1)( 6 k  1)
1 1   (1) k (k!) 2

.04659629166272193867...   2 F1  2,2, ,   
2 4  k 1 (2k  2)!

.04655015894144553735... 
2 .04662202447274064617... 
 2


 log 2 
2  12

1
=

0
.04667385288586553755... 
2
 /2
 log sin x  dx 
2
log 2 x dx
GR 4.261.9
1  x2
11
4 
e
GR 4.224.7
0
( 1) k 1 k 2

k  0 k !( k  2 )

5 log 2  (1) k 1



.04672871759112934131... 
18
3
k 1 3k  9
.046748650147269892437... 
 csc ( 1)   csc ( 1)
1/ 4


 12k

k 1
2
1
 30k  18
3/ 4
2
Hk

5
k 1 ( k  1)

 (k )
.046842645670287489702...   k 2 
k 2  k

1
1 .046854285756527732494...  

k  2 k ( k  1) log k
.04682145464761832657... 


 
=
m 2 k  2
m
1

log k
   ( x)  1 dx
k
 (k )


1
 2 

k
 ( k  2)! k !   k I 
k 2
.046943690640669461497... 


k 2
1 .04685900516324149721... 

 k
k 1
2

1
1
 1  2 arctan


2
2
2
 /4

0
sin 3 x
dx
1  cos2 x
sin 3  0 
 

3
3/ 
8 .0471895621705018730...  5 log 5 
.04704000268662240737... 


 2k 
(2 k  1)!!
1

  k


k
3
k  1 ( 2 k )!!4 ( 2 k  1)
k  1  k  16 ( 2 k  1)


1
1 
 1
1 .04719755119659774615... 
 arccos   ( 1) k 1 


 6k  5 6k  1
3
2
k 1

1
= 
k  0 ( k  1)( 2k  1)( 4k  1)
.04719755119659774615... 

1 
 2k 

=
k
k 0

0
.04732516031123848864... 
1 .04740958101077889502... 
.047430757278760508088... 
93648
.04747608302097371669... 
5 .04749726737091112617... 
1
  k  16 (2 k  1)

=
J328
dx

6
x 1


0
x 4 dx

x6  1


0
x1/ 2 dx
x3  1
1
 (i )   (i )
4

30( 4)
4
a(k )

  5 
93
31
k 1 k

k (2k )
1
2 2  3 3  
72
36k
k 1
10 


2

Titchmarsh 1.2.13
.04761864123807164496... 
.047619047619047619
=
.04794309684040571460... 
.04821311371171887295... 
.04828679513998632735... 
.048298943674391117963... 
.04831135561607547882... 
log 4 x
7 4

 9 (3)  
dx
( x  1) 4
3
90
1
1
21


1
1
5
  ( 3)   5


3
3
4
k 2 k  k
k 1 k ( k  3) ( k  2 )

1 
 1

log 2   (2)  1     2  log 1  2   

k 
k 2  k
log 2 1
 
4
8

1

k 1
k 1
2 1

(k!) 2
 

18 2 k 1 (2k  2)!
 
=
1
k
4k 2  6k  2

 2k 
k 1
 
k 

log x dx
5
1
 x2

k3

6
k 1 ( k  1)
.04852740547184035013... 
( 3)  3 (5)  3 ( 4)  ( 6) 
.04855875496950706743... 
24 log 2  3 2  41
   log(1  x ) x 2 log x dx
108
0
1
.04871474579978266535... 
.04878973224511449673... 
1 .04884368944513054097... 

3 cos1  sin 1
k4

  ( 1) k 1
( 2 k  1)!
16
k 1

2035
log 2 x
2 (3) 
  6
dx
864
x  x5
1
( 2 )   ( 3) 

3
tanh
8
 64 log 2 
3
3
222 .04891426023005682...

 3
2


k
k 1
5

x
0
 3/ 2
Li2 ( x ) 2 dx
k


 2 (k
k 2
(1) k  (k )  1

2k (k  1)
k 2

 
x
 (2 k )  1
k 2
 (4k  2)(4k  4)   2 (8k  8)
3  19 log 2
 
 2 log 3  3 (1) 
2 4
2





1 1 
i
i 
1

   1      1      3

 2
5 2   2
k 2 4k  k

k 1  ( 2 k  1)  1
 
=  ( 1)
4k
k 1


1
 3 ( k )
1 .0483657685257442981...   k


3
2k
k 1 (2  1)k
k 1

1  (1)  2 
( 1) k
(1)  7  
.04848228658358495813... 
         
2 
 3
 6 
36 
k  0 ( 3k  4 )
.04832930767207351026... 


2
1
 k4  k3
k
1
2
 1) k

.04904787316741500727... 

k
 ( k  1)( k  2)
 ( 2 )  2 ( 3)  4 
k 1

=
 ( 1)
k 1
k 1
1 .04904787316741500727... 
 ( 1)
k 1
k 3
3 .04904787316741500727... 
4 .04904787316741500727... 
=
.04908738521234051935... 
5 .049208891519142449724... 
.0493061443340548457...

=
.049428148719873179229... 
k 2  ( k  2)  1
( 2 )  2 ( 3)  3 

=
9 k 2  11k  4

k 2 ( k  1) 3
k 2

k 2  ( k )  1

Hk 1
2
k 1 k

k 1
 (2)  2 (3)  
3 
k  2 ( k  1)

H k H k 1

k 1 k ( k  1)
 ( 2 )  2 ( 3)  1 

64

( 1)
1/ 4
csc  ( 1)
1/ 4
k 1

 27 k
k 1
 i csc  ( 1)
3/ 4

1
 3k
3
( 1) k
  4
k 2 k 1

1 2
 1 
 7  6 3 log 3  27 log 2 3  12 (1)    
 3 
8

Hk
 
= 

k  1 k ( k  1 / 3)



k 1
1
log n k
.049472209004795786899...  1   ( 4)   4
 
 k4
k  2 k log k
n 1 ( n  1)!k  2

263 
( 1) k 1
.04952247152318661102... 

 
315 4
k 1 2 k  9
2 .04945101468877368602... 

2
x 2 dx
0 ( x 4  4) 2

49 e
k4
 
k
16
k 0 k !2

log 3 1
 ( 2 k  1)

 

2
2
32 k 1
k 1

dx
2 x 4  x 2
4

 k  ( k  1)  1



3
sech 3
1
 3 3
2
e e

9
( 1) k k 4  2 k 
  k
.04971844555217912281... 
 
128 2
k  1 4 ( 2 k  1)  k 
.04966396370971660391... 
3 7
.04972865765782647927...  3 log 

2 6

 .04974894456184419506... 
.04976039488335674102... 

 
=
.04978706836786394298... 
.04979482660943125711... 
.04984487268884331340... 

2
.04987831118433198057... 

2
4
k 1

12
 1 1
 Lik     
  3 3

4
 3 log 2
1


5
k  2 k log k
2
k 1
log(1  x )
dx
(1  x) 2
0
1
 2 log 2 

 (1)
1 5

6
2
 3   5 5  1 
1
HypPFQ 1, , 2,  , ,   
18
 2   2 2  4 
1
e3

( 1) k 1 k 2
23
14 
 
k
e
k 1 ( k  2 )! 2

5 log 2 19
1

  k
6
36
k 1 2 k ( k  2 )( k  3)
1

.04987030359909703846... 

(1) k 1
 k 2  4k  3 
(1) k 1 (k!) 2

k 1 ( 2k  1)!( 2k  1)

  ( x)  1dx
5
 2  4 log 2 
( 1) k 1 k


2
k 1 ( k  1) ( k  2 )


.05000000000000000000

1
9

=
  1  2 
k 
20 k  4 

k ( k  6)
= 
2
k 1 ( k  3)
1 .05007513580866397878... 
.05008570429831642856... 
.05023803170814624818... 
.05024420891858541301... 
J1061
21/ 2 31/ 4  1/ 2
CFG F5
 ( 3)
24

51  2  ( 3)
1


 
3
48 24
2
k 1 k ( k  2 )

3 ( 3)
( 1) k 1 k 2
( 2 )  log 2 
 
3
4
k 1 ( k  1)

2 .05030233543049796872... 
 1  log (k ) 
k 2
2

 .05037557702545246723... 
.05040224019441599976... 

12

log 2 2   (2) log 2 

6


.05051422578989857135... 
.05066059182116888572... 
1 .05069508821695116493... 
=
.05072856997918023824... 
2 .05095654798327076040... 
1 .05108978836728755075... 
k 2
(1) k log 2 k
k2



1
 (2 k  1)
k 1
.0504298428984897677...
2


1 i 3
1 i 3
1

  1 i 3  
 1 i 3  
2
 4 
 4 

=
 (2)
8  2G 
 (3)  1
4

2
3
1

2
4



1
1
 3 log 2   log(1  x 4 ) log x dx
0
2
log x
dx 
x5  x 3

x 2 dx
0 e2 x (e2 x  1)
1
22
1 (1)  1 
1  (1)  1 
 3 
     (1)    
   
25
100 
5
 10 
 5 
2 2
1
1
 9

  (1 )  
 5
125 16 25

1
I 4 (2)  
k  0 k !( k  4 )!
2

sinh





1
 (5k  4)
k 1
2
LY 6.114
1 
2

 1  2k
2
k 1
1/ 
1
e  e 1/ 
cosh 

2


1
 (2 k )! 
k 0
2k
AS 4.5.63

2 .05120180057137778842... 
.051297627469135685503... 
21/ k

5
k 1 k

1
21 (3)  4 2 log 2  16 G
128

 /4
=
 x log cos x dx

0
.05138888888888888888

=
 .05140418958900707614... 
.0514373600236584036...

1  dx
2  13 
x


1
2
192
log x dx
x5  x
1


  3 log 2

2  16



2
1
     x 2 e  2 x log x dx
16 8 
0

4 9
( 1) k 1

  k

3 4
k 1 3 k ( k  1)( k  2 )
2
3
1 .05146222423826721205...  2
 csc 
5
5 5

7
k 2  4k  1
 .05153205015748987317...   ( 2 )  6 ( 3)  6 ( 4 ) 
 

4
8
k  2 ( k  1)
 
=
 ( 1)
k 1
H
k
k 1

1 .0515363785357774114...

( 3)
k 3  ( k  1)  1 
Berndt 5.8.5
(( k  1)  1) 
k 1

k (2k  1)
1  (1)  4 
6 
     (1)     
20 
25k
5 
5
k 1

13700
k
.05165595124019414132...  5040 
  ( 1) k 1

e
k !( k  7)
k 1
.05160484660389605576... 




GR 4.355.1
24 log



 log1  x

.05145657961424741951... 


37

720
2 .05165596774770009480... 
 
2
4
csc 2



2 2 2

1
= 

4
2
1
k 1 2 k  2 k  2
7 ( 3)
  ( 3) 
1 .05179979026464499973... 
8
 /2
 x log tan x dx
=
0


 1
2

k ( 2 k  2)


2k
k 1
cot
1


3
k  0 ( 2 k  1)

2
Hk


k
k 1 2 k
2





 .05181848773650995348... 
=
2 .05226141406559553132... 
 .05226342542957716435... 

 .05230604277964325414... 


5

96
 4 (3)

 (6)

 .052631578947368421
k 1
( k )
k 2
6k


2
0

k3
Titchmarsh 1.7.10

sin 2 k

k!
k 1

si (1)  sin 1
(1) k 1 k

 
2
k 1 ( 2k  1)!( 2k  1)

k


k
k 2
)
k 1 2 (1  2
1
=

19



jk



jk
j 1 k 1 2
k0 ( k )

2k
k 1



1
10
1
1

log
 arctanh
  2 k 1
( 2 k  1)
2
9
19
k  0 19

1
 1 log 2
.05274561989207116079... 
 
 

8 6
4
k 1 ( 4 k  2 )( 4 k  3)
 .05268025782891315061... 
.052878790640516702047... 



 ( k )
e cos3 sin(sin 3) 





 3   5 5  1 
(1) k 1 k!
1

HypPFQ 1, ,  , ,    
18
k 1 ( 2k  1)!( 2 k  1)
 2   2 2  4 


1
1
1
 .05239569649125595197... 





3
3k
e 1
k 1 e
k 1 cosh( 3k )  sinh( 3k )

5  6 log 2
x dx
  log
 
 .05256980729002050897... 
16
x  1 x3
2
4 .0525776663698564658...


 .05238520628289183021... 


J385
( 1) k 1


k 1 k ( k  2 )( k  4 )
 2
3
x sin 2 x dx
4 .05224025292035358157... 

 2  
3
1  cos x
0

log 3 log 2

1
 .05225229129512139667... 


 

2
4
3
4 3
k 1 36 k  6 k

dx
 
=  6

5
4
x  x  x  x3  x2  x
1
 .05208333333333333333



178
1
1
 (2 k  1)!!
   


225 5
k  1  ( 2 k )!!  2 k  5
 .05290718550287165801... 
1
4 2

3

35 (3)  16G  2
2
  2  ((k2k)1)
log 2 

k 1

log 3 log 2
 ( 2 k  1)

 
12
3
4k
k 1
4k
K148

4
1
  3

3
k 2 4 k  k

1
=  k

k 1 2 ( 4k  12)
.05296102778655728550... 
2 log 2 

 

 .05296717050275408242...  1 

 .05298055208215036543... 
1 .053029287545514884562... 
74

720

 ( 2 k  1)  1
k 1
4k


( 1) k

4
k 0 ( k  2)

3
log 2 x
G 2
dx 
32
( x  1) 2
1

2
16 2  2



2
64
csc 2

8



k 1

1
  (8k  1)

2

1

(8k  7) 2 

 .05305164769729844526... 
6 .0530651531362397781...

 .0530853547393914281...

 
1
6

2
( 2 k )!! 2 k
 

k
3
k 1 ( 2 k  1)!! 3

3 ( 3) 7
( 1) k



 2
3
2
4
k  2 ( k  1)

1 2
log 2 x dx
x log 2 x dx

=  4
0 1  x
x  x3
1

2  3 2 arcsin

1 .053183503816656749...


  ( jk  1)  1 
j  3 k 1

1 .053252407623515317...

8 log 2  2 ( 2 )   ( 3) 

 2k
k 1






 .05357600055545395029... 

   ( jk  5)  1 
j 1 k 1


 .05358069953485240581... 
1
 ( k  3)! k ! 
k 1
.05361522790552251769... 

128


 .05362304129893233991... 

csc 2
 
 
   2 2 sin

4 2
2 2
 ( k )  12 
k3
k !!
 .05362963066204526106...  

k  0 ( k  4 )!

1

2 .05363698714066007036...  
k 1 Fk  1
k 2




 ( k  3)
k 1
2k


log 2 1  i 3  7  i 3  1  i 3  7  i 3 
 


 
 


6
3
12
4
12
4





1
 
= 
3
k 1 ( 2 k  1)  1


 (2 k )
2 .05339577633806633883...  2 log   log 2  2 log csc
  k 1 
k
2
k 1 2

1 

 
=  2 log 1 


2k 2 
k 1

1
.053500905006153397...
 

k  0 ( k  4 )! k !

1  (1)  2 
log x
(1)  9  
dx 
 .05352816631052393921... 




 
 


 5
 10  
x5  1
100 
1
 .05330017034343276914... 


1

4
 k3
5 .0536379309867527384...


 x( x  1) ( x)  1 dx
2









1
5 1 3 1
1 5 1

(1  i )   Ei ,    1F1  , ,    k
 4 4  4 4
 4 4 4  k  0 k!4 (4k  1)

 2  7( 3)
1
k

 .05391334289872537678... 

 
3
16
27
k  2 ( 2 k  1)
43867
3 .05395433027011974380 =
   ( 17 ) 
14364

log 3 1
log x
dx 
 .0539932027501803781... 

 
8
12
( x  2) 3
1
1 .0536825183947192278...

1 .05409883108112328648... 
3
3
 1  log  
2
4

kHk
k 
k 1 3


  (1   ) 
1 .05418751850940907596...     1 
    k ( k ) 



k 2

5471
1
 .054275793650793650
=
 

100800
k 1 k ( k  3)( k  8)


2
 (k )   (k  1)
.054298855026038404294... 


 log 3  
3k
3 3 18
k 2
2

.054612871757973595805... 
.054617753653219488...

6 .0546468656033535950...


 2 log 
4

7 (3) 1
 
8
8 2
1
 1
 log 1   
 e
e
3

3
 2 log 2 

 ( 2k )

 2k (2k  2)4
k 1
k
dx

x
 1)
 e (e
x
1
1  (1)  7 
 2
     (1)    
 6
 3 
12 
1


 
=
1 .05486710061482057896... 





1

  log 1  3  log x dx 

x 
0

 1
2  1

( 2 k  1)!!
 (2 k )!! 
k 1
2k

( 1) k 1


k 1 2 k  8

 (k )
1 .05491125747421709121...   (5) ( 6)   1 6 
k
k 1


1
H ( 2) k
 1
1 .05501887719852386306...  5 Li2     k

2  
k
 5
k  0 5 ( k  1)
k 1 5

4
k
 .0550980286498659683...  28 log  8   k

3
k  0 4 ( k  1)( k  3)
 .05490692361330598804... 
log 2
7


2
24
J166

HW Thm. 290

 .05515890003816289835... 
log 2

4
 .0551766951906664843...


2
0
1
dx

2
 4 log x 1  x 2
 (k )  1


1

1 .05522743812929880804... 
k 2
 ( k  1)!   k e
 (2 k )



 


k 1

1
 2
k I1    1 

 k
2k 
 k !( k  1)!   
k 2


2
GR 4.282.7
1/ k 2
1  1 
k 1

5
( 1) k ( 2 k )! k 3
 .05524271728019902534... 
 
( k !) 2 4 k
64 2
k 1

 2  6 ( 3)
k
 .05535964546107901888... 

 
3
48
k 1 ( 2 k  2 )
9 .0553851381374166266... 
82 


1 .05538953448087917898... 
 2
7
coth  2  
2
6
3

32 2
3

2 .05544517187371713576... 
1

 
=

 .0554563517369943079...


 .05555555555555555555
=

1 .05563493972360084358... 

 

 .05566823333919918611... 
=
.055785887828552438942... 
2 .05583811130112521475... 




.0558656982586571151...

2
log x
dx 
4
1
0
x
2


2
k 2 k  2

 ( 1)
k 1
k 1
2 k  (2 k )  1 
2
log x
dx 
2
 x 2
x
0

x 1 2
log x dx 
4
1
x
0


2
GR 4.61.7

( 1) k 1 k 2  2 k 
11
2
  k
 
4 2
k  1 4 ( k  1)  k 
1

18
1
cos 2  cosh 2  2 sin 2  2 sinh 2 
2

4k


k  0 ( 4 k )!( 2 k  1)

( 1) k
1
2

csch 3   2
18
9
k 0 k  9

 1  H ( n) k 
1
5
log       k  
4
4
4 k 1 5 
n 1 

Hk


k
k 1 2  1

1


k
k 2 4 ( k )

38 .05594559842663329504...  14 e 





1 .05607186782993928953... 

1 1
1

cosh  e1 / 3  e 1 / 3  
k
3 2
k  0 ( 2k )!9
2 .05616758356028304559... 
5 2

24

log(1  x 2 )
0 x(1  x) dx 
k 1


5
k  2 k log k

.05633047284042820302... 

.05639880925494618472... 
5
  ( x)  1 dx 
4
1 8 log 2 2  8 log 2 log 5  2 log 2 5  4 Li2
1
 1
 1  4 Li2    
5
 4
( 1) k 1

k
2
k 1 4 ( k  1)
1
x log x

dx 
0 x4


 
=
=
.05650534508283039223... 
1 .05661186908589277541... 
=


e 1
4

x tan x
dx
2
4
x
0
GR 3.749.1
3
3
3 1
3
3
1
 1
log  log 2  Li2    log  Li2   
2 2
2
2
2
2
 3
 2

H k (k  1)

k 1
3k k
i i i 2
i


 log 2  Li2 (e 2i )    log( x sin x ) dx 
2
2
12
2
0
1

2 .056720205991584933132... 
1
73 .05681827550182792733... 
3 4
4



2
coth  
csch 2  
3

4

1

4
4
 1) 2
k 2


3
( 1) k
( 2 k  3)!
.05685281944005469058... 
 log 2  

 
4
( 2 k )!
k 1 ( k  1)( k  3)
k 2

1
 
= 

k 1 2 k ( 2 k  1)( 2 k  2 )
.05683697542290869392... 
 (k
k 1
 (2k )  1
k 1 k

( 1) k 1 k 2
8
.05696447062846142723...  3 
 
e
k 1 ( k  2 )!

1 


( 1) k
.05699273625446670926... 
 csch sech   4
8 8
2
2
k 2 k  1

1
1
.05712283631136784893... 

  ( 2 )   ( 3)   4
3
2
k 2 k  k

=

2
J249


 
=

 ( 1)  ( k  3)  1   dz( k ,2)
k 1
k 1
k 1

7  675

120
4
.05719697698347550546... 
.05724784963607618844... 
1 .05725087537572851457... 

G

16
e
4x
0
3
log x
4
 x3
x
1
x dx
 e4 x

1
 Ei (1)  Ei ( 1) 
2
1
 (2 k  1)!(2 k  1)
1
=
sinh x
SinhIntegral (1)  
dx 
x
0

 1  dx
x
 sinh x 
1

7 2 ( 3)
r ( n)
  3
5 .0572927945177198187... 
2
k 1 n
 ( 3)
74
2
.0573369014109454687... 
 2 log 2 

4 
4
720 12
.05754027657865082526... 




1 .0578799592559688775...



4

cos 4 x
dx
2
x

1



17 .05777785336906047201... 
 .05796575782920622441... 
GR 3.749.2
 
=


log 2 (1  x )
0 1  x dx 
0 F1  ;1; 5
I 0 (2 5 ) 
7(6) (3)2


4
2
log 2 
 
2
2
 ( 1)
cos k 3

3 
k 1 k
1
log 4 2

4


( 1) k


5
4
k 2 k  k



e

14
k5
 .0576131687242798354... 
  ( 1) k 1 k 
243
2
k 1
1
1 .05764667095646375314... 
Li3  ei / 3   Li3  e  i / 3  
2
 .05770877464577086297... 
GR 8.432.1
k 0
k 1



k 1
2
k 1

5k
 
2
k  0 ( k!)
Hk

k5
 (2 k  1)
2 k 1

(2 k  1)
Berndt 9.9.5


1
1
  arctanh 2 k  2 k  
k 1
 ( k  1) 
k 1


 
=
log log x dx

(1  x ) 2
1



 .05800856534682915479... 

k 2

 .05807820472492238243... 
 (k )
k
5
GR 4.325.3

  (5)  1  
3 7


2 6
k 2

( 2 k  1)!!
 (2 k )! 3
k 2
k
 (k )  1
k
5



 1
1
  Li  k   k  
k 1
5
.058086078279268574188... 
 /4
=

0


 .05815226940437519841... 


sin 3 x
dx 
1  sin 2 x
3 ( 3)
23

2 .0581944359406266129...
1 
 1  i
 1  i
 1  2 arccoth 2  i arctan 1 
 
  i arctan 1 


2
2
2 
1

 1 ( k )
Berndt Ch. 5

k!
3 2
 1
.058365351918698456359...  2 Li2     4 log 

2 3
 2
k 1
(1) k 1 k 2

k
2
k 1 2 ( k  1)



1 .05843511582378211213... 
 1 5 1
2 F1  1, , ,  
 4 4 4

.05856382356485817586... 


4
k 0
k
1

(4 k  1)
 ( 2 k  1)
2k  1


x cot x
  2
dx
4
e 1
x 4
0
k 1
.05861382625420994135... 
4 .05871212641676821819... 
.0588235294117647
=
.05882694094725862925... 
4
GR 3.749.2
, volume of unit 4-sphere
24
1
1
 sin 2 arctan 
17
4
 (4)

 2 ( 4)
cos 4 x
dx
ex
0
 (1) k 2 4( k 1)  
k 0

 (k ) log k
k 1
k4





 .05886052973059857964... 


 .05889151782819172727... 




1
1
2
1
1
 log 
arctan

18 9
3 9 2
2
1
9
1
log  arctanh

2
8
17
 ( 1)
k 1

 17
k 0
k 1
2 k 1
kH 2 k 1

2k
1

( 2 k  1)

Hk sin k

k
k 1
 3 

 .05897703250591215633...  log    
 4 4
1 .058953464852310349...



 (k )
4
k 2
k
1 
 1

  
 log 1    
 4k  
k
k 1  4 k


 
3 .05898844426198225439...  3 (5)   4 
  (3)  6 log 2 2 
6

2
1
1
Hk ( k  1)  1
 2  3  4 
2k  1  k
k
k 
k 1






7 4
( 1) k 1
 .05918955184357786985... 
 
  Re Li4 (i ) 
4
11520
k 1 ( 2 k )
 1 
1
 .05936574836539082147... 

 
8 3 k 1 (2k  1)(2k  1)(2k  3)

1
= 

2
k 1 16k  16k  3
1 .05940740534357614454... 

3 .05940740534357614454... 

4 .05940740534357614454... 
.059568942236683030025... 
e2
e
 1 
erf 
 
 2
2

J240, J627
1
 k !! 
k 2

e
1
k!!
 1 
  
erf 

2
 2  k  0 k!! k 1 k!

e
k
 1 

e 1
erf 


 2
2
k  0 k !!

 1 1 
1
 (k )  1
   2     ( 1) k

 

k
k 2

e
 3

9 log 3
1
 4 log 2  

4
4
k  0 ( k  1)( 2 k  1)( 3k  1)

log 2
3
(1) k
.059705906160195358363... 
 ( 3)    ( 3)   3 log k
4
4
k 1 k

43
1
.0597222222222222222222 =
 

720
k  2 k ( k  1)( k  4 )
1 .059638450439128955916... 




 .05973244312050459885... 
 .05977030443557374251... 
1 .05997431887...

2
16

 /2
(1  2 log 2  2 log  )  
10   ( 2) 
j4 
 x log x dx 
0
7 ( 4 )
3 ( 3)
 8 log 2 

8
2
( 1) k 1


2
4
k 1 k ( k  1)

J311
.06000000000000000000
=
4 .06015693855740995108... 
3
50
e  erf 1 

( 2 k )!!
 (2 k  1)!! k ! 
k 1
1

 
=

0
x
e dx
1 x

e  1
k!4k

1  e  erf 1     , 0,1  
2  2

k 1 ( 2k )!

5
1
1
( 1) k 1 k 2
.06017283993600197841... 

arctan
  k
18 2 2
2
k  0 2 ( 2 k  1)
5 .06015693855740995108... 
.06017573696145124717... 
.06034631805191526430... 
.06041521303050999709... 
1 .06042024132807165438... 

2123
1
 
35280
k 1 k ( k  3)( k  7 )

5 sin 1  6 cos1
k4
  ( 1) k
16
( 2 k )!
k 0

3
2
Hk
2  6 log  3 log 2   ( 1) k 1 k
3
2
2 ( k  1)( k  2 )
k 1
 
 
  1  1
 (ck )  1
   log
 c  

c 
k
1  n  n 
c  2 k 1
c2 n2 
112  9 2  24 log 2
  x 2 log(1  x 2 ) log x dx
108
0
1
.06053729682182486115... 
.06055913414121058628... 
3
512


 .060574229486305732161... 

k 1
 (k ) log k 315

 (3)
k   (k )
2 4

k0 ( k )
k
k 1 2  1

3
( 2 k  1)!!
1 .06066017177982128660... 
 1 

k
2 2
k 1 ( 2 k )!! 9


( 1) k 
=  1 

2 k  5
k 0 

5 .06058989694951351915... 

3 3
1


  k
2 4
k 1 3 k ( k  1)( k  2 )

(1) k 1
 
=  k
k 1 2 ( 2k  1)


1
1
k  1  ( 2 k  1)  1
.06097350783144797233...   ( 1)
    arctan 

2k  1
k
k 1
k 2 k
.060930216216328763956... 


2 log
CFG B4, J166

1
( 1) k 1
 
.061208719054813641942...  sin
2 k 1
4
k 1 ( 2 k )! 2
5
4
4 log 2
.061385355812633918024... 



27 81 3
81
2
( 1) k

2
2 
k  0 ( 3k  1) ( 3k  4 )

4 log 2

 3
 1  sech

3
3
2

( 1) k
 
=  4
k 2 k  k
1 9  6 2  24 2  40
.06154053831284500882... 


3
36  24 2
.06147271494065079891... 

.06158863958792450009... 
1 .06160803906011000253... 

 
=
.06160974642842427642... 
=

 .06163735046020626998... 
.061728395061728395
=
.06173140432470719352... 
.06177135864462191091... 
.06196773353931867016... 
1 .06202877524308307692... 

.06210770748126123903... 
1
304 
 
 
4
105 

( 1)

0
sin 2 x
dx
(1  sin x ) 2
k
 2k  9
k 0

2  (1) 
1 
1 
k

 1 

   (1) 1 
   
2
2
16 
2
2 


k 1 ( 2k  1)

k (2k  1)


2k 1
k 1
3

1/ 4
 
 csch 21/ 4 

3 / 4 csc 2
4 42

( 1) k


4
k 2 k  2

1
1
( 1) k 1
9  8 cos  4 sin  
k
2
2
k 1 ( 2 k )! 4 ( k  1)

5
k4
  ( 1) k 1 k
81
5
k 1
k

( 1)
 HypPFQ[{},{1, 1,1}, 1]

4
k  0 ( k !)

( 1) k 1
35  40 log 2  6 ( 3)   4

4
k 1 k ( k  1)

 2k  k 2
2 3
k 1
  ( 1)   k
k6
25 5
k 1

arccsch
2

4

3
16




1
log 2 x
dx 
x4  x2
65
1

.062346338241142771091... 

csch 2 2 
16 4 2
8 .0622577482985496524...
 /4
1

0
x 2 log 2 x
dx 
x2  1

( 1) k 1

2
k 1 k  8


.06238347847451615361... 
  ( k )  1 ( k  2)  1 
k 2
.062400643626808666895... 
.062500000000000000000 =
9450
 12
210 (8)    (4)    (kk) log k

4
k 2

1

16
k
k 2
3


 
=
 e
k 1


 
=
 (x
=
k

 (1) k

1  dx
4 9 
x
 log1  x
1


 
=
cos 2x
e 
2
0

1 .06250000000013113727... 

dx 
Prud. 2.5.38.9
kk
1
22
k

2

log 2 2
H
  k k1
12
2
k 1 2
4
11
28 2

 84  14 ( 3)  4 (5) 
.06269871149984998792...   ( 6) 
90
3

1
 
=  6
4
k 1 k ( k  1)
1 .06269354038321393057... 

1
2
1

4k
k 1 2
k 1

.062515258789062500054... 
x
1
k
k 1

.06251525878906250000... 
k 1
x dx

 2) 3
0
 

 ( k  1) (2k  1)  1 
2


k
1

5
(1  k  2 ) 2
4
2

2
1 .062714148932708557698... 
  ( x) log x dx
1
1
.062783030996353104594... 
 log(1  x
12
) dx
0

3
kH
3 2
2 2
  2kk 1
 log 2 
log
2
4
2
2 2
k 1
2
4


35
1
.06290376269772511922...  56 

 5 ( 3)   (5)   5
4
6
18
k 1 k ( k  1)

log 2 2
( 1) k 1  ( k  1)
.06310625275909953582... 
  log 2    

2
k 1
k 1
4 .0628229009806368496...

(1) k 1 log k
k (k  1)
k 1

.0632539690102041...

.063277280080096903577... 


1
G 19


   x 3 arccot x log x dx
32 4 72
0

10 2
1
 35   4
4
45
3
k 1 k ( k  1)

1
1
1
1 .06337350032394316468...  8  4 sinh  8 cosh  
k
2
2
k  0 ( 2 k )! 4 ( k  1)
.06332780438680511248... 
.0634173769752620034...

.0634363430819095293...

1 .06345983311722793077... 
.06348038092346547368... 
1 .06348337074132351926... 


=
.06364760900080611621... 
.06366197723675813431... 
.0636697649553711265...

4

4

1
1
 1
 ( 3)    
4 
 2
1536
1536
k  0 ( 4 k  2)


11
1
k
 2e  
 
2
k 1 ( k  3)!
k  0 k !( k  2 )( k  3)( k  4 )

8G 
1

Berndt Ch. 9
 log 2  3  
3
3
k  0  2k 
2
  (2 k  1)
k

1

k ( k  3)
k 1 2

1
 1
I0    
k 
2
 2
k  0 ( k !) 16

( k  12 )!
e 
2
k  0 ( k !)

1 2
( 1) k 1 k
arctan 
  k
2 5
k  0 4 ( 2 k  1)
1
5
log p
1  log k
(4)
 
  4


4
 ( 4 ) k 1 k
( 4)
p prime p  1



(k )
4
k 1 k


 

 .063827327695777400598... 
=

2 log 2 
2
12
21
( 4)
1 .06387242824454660016... 

2
 ( 6)
2
(1)  2 
3 .06387540935871740999...    
 3
12 .06387540935871740999... 
 1
 3
 (1)   

1

2
( 1) k

3
2
k 2 k  k


1 .06388710376241701175... 
1
k
k 1
4
7 ( 3)
log 2 
64
4

(1) k 1 0 (k )
.06400183190906028773...  
2k
k 1
 ( 3) (5)
1 .06404394196508864918... 
 2 (4)
.0639129291353270933...

.06407528461049067828... 
.06413507387794809562... 


Hk
 (2 k  1)
k 1
3
5

9
( 1) k ( 2 k )! k 3

( k !) 2 2 k
9 3
k 1
28log 
2
2 28arccsch2
.064205801387968452356... 




125 125 5
125
125 5

k3
k
 
=  ( 1)
 2k 
k 1
 
k 
.06415002990995841828... 


2k
.06422229148740887296... 
1
20 


5 2
  ( 4 )  2 ( 3) 
3

.064274370716884653601... 
7 log 2  2 log    
13

2

k
k 1

3
1

( k  1) 4
(1) k 1 k 2
 (k  1)(k  2)  (k  1)  1
k 1
log 2 1
( 1)
  

3
6
k  0 3k  9

1  dx

=  log 1  6  7

x x
1

k
.06438239351998176981... 

 

1 ( 3)  2 
1
.06446776880150635838... 
    
4
 3
486
k  0 ( 3k  2)

k ( k )
.064483605572602798069...   ( 1)

2k
k 2

1
1
 (2 k )  1
.06451797439929041936... 

 2
  ( 1) k 1
2
e 1  1
 2k
k 1


.064710682787920926699... 

 4  2 2  1 
.06471696327987555495... 
x 2 log x
4
dx
 2  2 Ei ( 1)  3   
e
ex
0
4

x 2 (1  x) 2
0 x 8  1 dx
1
1 .06473417104350337039... 

 

 
=
1  (1)  1
 3
 5
 7
     (1)     (1)     (1)    
 8
 8
 8
 8
64 
1  ( 1)  1 
2
(1)  3 

        
 8
 8 
32 
8

 (1) k 1
(1) k 1 
(1) k / 2 






2

(4k  1) 2  k  0 (2k  1) 2
k 1  ( 4k  3)
1
1
3 log 2
log(1  x 3 )

.06487901723815465273... 
dx


 
2 3 3
2
(1  x) 2
0

=
.06498017172668905180... 
.06505432440650839036... 
.06505816136788066186... 
.06506062829357551254... 
.06514255850624644549... 
4
6
k2 1
( 4 )   ( 6) 

  6
90 945
k
k 1

Ei (1)  1
 ( k  1)
 1   ( 1) k
e
( k  1)!
k 1
2

log k
   ( 4)

k4
k 1


11  3 log 3
1
( k )  1


  2
  ( 1) k
12
18
2
3k
k  2 9 k  3k
k 2
log 2 2
3 log 2
 2  1  2  1  1   (3)
log 
Li2     Li3     Li3    

 3  2  3  2  3
4
5
2
16

log 2 (1  x )
dx
= 
x ( x  4)
0
1

 
2

k
2
( k  2) 2
k 1

2
(2 k )!!
1 .06519779945532069058...  6 
 
3
2
3
k  2 ( 2 k  1)!! ( k  k )
.06519779945532069058... 
5

.06528371538988535272... 
G 


2 8
 ( k  1)
2
( 1) k

2 
k  1 ( 2 k  1)


log x
dx 
2
 1) 2
 (x
1

.06529212022186135839... 
.06530659712633423604... 
.06537259256...

.06544513657702433167... 
.065487862384390902356... 
.06549676183045346904... 
sin 2 ( k / 2)
1
 1
k 1
log sec    ( 1)

 2  k 1
k
2

10
( 1) k k
6  
k
e
k 1 ( k  2 )! 2

( 1) k log 2 k

k
k 2

241 8 log 2 log 2 2
(1) k 1 H k



144
3
2
k 5
k 1

1  (1)   
(1) k
(1)  1    
     
   
2
4 
 2   k  0 (k   )
2
J523
1 
2  311 / 6 

1/ 3

 1  


3  3i  2    

 3  




 
=
 (1)

k 1


 31 / 3  i31 / 6 
  2 3 1  31 / 3
3  3i   2 
2


 (3k )  1
k
3
k 1
.06557201756856993666... 
=
.06567965740448090483... 
.06573748689154031248... 
.065774170245478381098... 
.06579736267392905746... 
.0658223444747044061...

.06598267284983230587... 
.06598587683871048216... 
x
.06598803584531253708... 
e e 
6
1
.06604212644480172199... 
.0660913464480888534...
.0661733074232325082...



k 2
1
3k  1
3
dx
 x  x 4  x3
5
1
3 arcsin
1 
3
2
20
csc

5


4 5
( 2 k  1)!!

 (2 k )!!3 (2 k  1)
k
k 1
cot

5

1

2
2 ( 3)   ( 4 )  4 ( 2 )  10 


 (5k
k 1
k
k 1
.06620909207331664583... 
=
.06633268955336798335... 

9 3
81log 3
 4

  2   (3)  3 (1)  
 3
2
2

1

3
2
k 1 k ( 3k  1)

2 log 2 19
( 1) k 1

 
3
36
k 1 k ( k  2 )( k  3)
7 ( 3)
128

 (2 k )  1
(  1)(  1) csc 1
 
log
2

 2k k
k 1

1
2
( 2)

 

2
150
25
k 1 (5k )

2
5
Hk

 
36 24
k 1 k ( k  1)( k  2)( k  3)
2

19
2k
e

 
16 48
k  0 ( k  4 )!

log 2  1
1
  

2
4
8 2 k 1 16k  12k  2
=
 


 
81 



2


1
J840
1
( k  1) 4
csc 1
2
2
2  (  1) cos1  ( 3   ) sin 1 
2  2
cot 1   (2k )  1
 2 3



2(  1)(  1)
2
 2k
k 1
1  ( 3)  3 
( 3)  1  
6
        
 4
 4
256 
1

 1) 2
2

1
k 
k 1
log 3 x
x4  x2
2
2
1

.06637623973474297119... 
.06640625023283064365... 
 9
log   
 10 
1

22
p prime

p



1
(2 k )
( 4 k  2)




k



k
4k 2
2
1
k 1 4
k 1 16  1
k 1 4

2 3
( 1) k ( 2 k  1)!!
 .06649658092772603273... 

 
3 4
( 2 k )! 2 k
k 2

1
.06642150902189314371... 

.06660601672682232541... 
e
k 1
ek

log 2 2
log 2 (1  x )
.06662631248095397825...  1  log 2 

dx 
2
(1  x ) 2
0
1

2 .066646848615237039218... 

.066666666666666666666 =
1 
5
 11  
116 3   ( 2)     ( 2)   
1728 
 12 
 12  
1
15
k ( k  6)

=
 ( k  2)( k  4)
J1061
k 1

 2k 
16
1
 1
   3,    k
  
 2
15
k  0 4 ( k  3)  k 
1
8
1
log  arctanh 
.06676569631226131157... 
2
7
15

1
 
=  2 k 1

( 2 k  1)
k  0 15


 (3)
log 2 x
x 2 dx
.06678093906442190474... 
  4
dx   3 x
18
x x
e 1
1
0
1 .066666666666666666666 =




1 

2k 
 1 2
k 2
k 1 ( k )
3k
k 1

1
1 .06687278808178024267...   k  2
k 1 k

1 .06683721243768533963... 
.06687615866565699453... 
3 .06696274638750199199... 


 2k 
1
2 log 2  3  4 3  4 log 2  7     k
k  1  k  16 k ( k  1)

2 2 log 2 2 log 3 2
 1

 4 Li3     3 (3) 
 2
3
3
1

 
=

log 2 x
0 ( x  12)( x  1) dx
K148
2 .0670851120199880117...


3

15

k  (2 k )  1
csc2 1 cot 1
2
.067091633096215553429... 

 2


2
4
4
(  1)
 2k
k 1
i

1 .06711089410508528045... 
Ei e  i   Ei ei   1 
2

i
sin k
i
i
 
=   1  0,  e   0,  e   

2
k 1 k!k
.067519906888675983757... 
=
12 .06753397645603743945... 
1  5 
2 
 
arc cot(1  5 )  arc coth 2  arctan

5 
4



(1) k 1 Fk Fk 1

4 k (2k  1)
k 1
4G   
3
8
1
 2 log 2 

1
2
2  log x dx

 log1  x
0

.067592592592592952592 =
.06759686701139501638... 
.0676183320811419985...

10 .06766199577776584195... 
.067678158936930728162... 
169
1
 
2700
k 1 k ( k  3)( k  6)


1


1
(2 k )

cot  


2
k
2 10
5
k 1 25k  1
k 1 25

90
k2
  ( 1) k 1 k
1331
10
k 1

1
9k
cosh 3  ( e 3  e 3 )  

AS 4.5.63
2
k  0 ( 2 k )!



1 
5
11 2  6 H   , 2   9 log 3  12 log 2   3 3 (log 432  6)  


72 
  6 




 
=
Hk
 (6k  1)(6k  1)
k 1

( 1) k 1 k  ( 2 k  1)
16k



2
2
k
16
k 1
k 1 (16 k  1)


 ( k  1)
k2
1 .06771840194669357587...   ( 2)    
 ( k  1)  1  

k 1 k  1
k  1 k ( k  1)

.06770937508392288924... 
.06775373784985458697... 
.06788026407514834638... 
.06797300991731304833... 
 e
2e
2

2


2e
1



cos(2 log x )
dx
1  x2
0

cos  t
 1 t
2
dt 
AS 4.3.146
0
1 
 2 ( 1) 
6

 (k )  1
 ( k  1)( k  2)
k 2
Adamchik-Srivastava 2.22
.06804138174397716939... 
.06808536722795788835... 
1
( 1) k 1 k 2  2 k 
 

8k
k
k 0


6 6

 
=
dx
 x( x  1)
3
e
2x
1
dx
 e2 x

( 1) k  1 H k


k  1 ( k  1)( k  3)

5
.06814718055994530941...  log 2 

8


1
1  e 2
log

4
1  e 2
x
2
4
dx

 x3


1
1 .06821393681276108077... 
ee
e


1
 k !e
ke
0

2 .06822709436512760885... 
  (k )  1H
k 2
.06826001937964526234... 
.06830988618379067154... 
.06834108969889116766... 
2
6


2
4

4
2 k 1
coth  
1
 (
2
0
7  3 log 2

72


 ( 1)  (2 k )  1
1

4
2 
k 1 k  k
k
k 2
dx

 log 2 x )( x 2  1)
GR 4.282.3
2
log x
dx
x4
1


919
1
 
13440
k 1 k ( k  2 )( k  8)

( k )
.06838993007841232002...  
k
k 2 2

e1/ e  e 1/ e
1
1
1 .06843424428255624376...  cosh 
 
2k
e
2
k  0 ( 2 k )! e
 1 i 3 
 1 i 3 
8 (3)
  Li3 

  Li3 
1 .06849502503075047591... 



9
2
2




1
3
log(1  x )
 1
 1
.06853308726322481314...  log log 2  Li2     Li2     
 4
 2
2
x 5
0
.06837797619047619048... 
.0685333821022915429...


sin  3

3 

  1  2  
k 
k  2


k 2  2k  2
2
 2k  1
2 3
k 1 k
k

2
2 .0687365065558226299...  
 2 HypPFQ {1,1, 1,1,1,1},{2 , 2 , 2 , 2 , 2 , 2}, 2
5
k 1 k ! k

log k
.06891126589612537985...    ( 4)   4 
k 1 k
1 .068959332115595113425... 
2
5
2
4

 csc

5
5
5 5

x
0
dx
1
5
.0689612184801364854...
1 .0690449676496975387...


1 .06918195449309724743... 
.0691955458920286028...

8 4
1 
729
1
k 1
2
 1
7
2


  (3k  1)

4
1

4
(3k  1) 
( 2 k  1)!!

 (2 k )!!8
k 1
k
3
29



log 2
log x sin x
 
 
dx
2 8
4
ex
0
log 2
7
.069210168362720747484...  
 (3)   (3) 
8
8
log(2 k  1)
3 
k  0 ( 2 k  1)


Prud. 5.5.1.5
12 log 2  3  14
   x 2 arcsin x log x dx

54
0
1
.0693063727312726561...
.06934213137376400583... 
1  ( 2)  7 
( 2 )  3 
        
 8
 8 
512 

log 2 x
1 x 4  1
.06935686230685420356... 




1 
31 / 3  i31 / 6 

 


i
3
3
2


2  311 / 6 
2





31 / 3  i31 / 6 
  2 3 2  31 / 3
3  3i   2 
2




 ( 3k )  1
1
 
= 
  3
k
3
k 1
k  2 3k  1

1
.069397608859770623540...  
 HypPFQ{1,1,1,1},{2, 2, 2, 2},1
3
k 1 k!k

.06942004590872447261... 

32 2



x 2 dx
0 ( x 2  2)3 
2

1
 4
.0694398829909612926...   3  9 log 3 
 18       2
2
 3
6
k  1 k ( 3k  1)

1
 

k (2k )

5(5  2 5 )   
.06946694693495230181... 

 
50 
25k
5 2

k 1
.06955957164158097295... 
=
.069628025046703042819... 
.06973319205204841124... 
(1)
1  1  4 log 3 2  (3)
1
log 2 2 log 3  log 2 Li2    Li3   

2  4
3
4
 4
2
log 2 x
1 x  1 dx
3 7
G 




2 8
32 4
2 3 1
 
27
3

 (x
1
2
1
 x log
2
0
dx
 x  1) 2
x arctan x dx 
 

1 .06973319205204841124... 
2 3 2


27
3
=
k 0

 
k
1 
3 1
, 
2 F1  2,2,
2 
2 4


5
log 2
( 1) k
( k )  1

 
  ( 1) k
12
2
2 k 1
k 0 2 k  8
k 2


1
1



2
k
2
k 1 8k  20k  12
k 1 2 (3k  6k )

=
k
  2k 
3
3 .06998012383946546544... 
.07009307638669401196... 


1
dx
 log1  x  ( x  1)
3
1
.07015057996275895686... 
7 4

9720
 (k )

.07023557414777806495... 
 (2 k )!


1
log 3 x
dx
x4  x

k 2

12 .070346316389634502865... 
.0704887501485211468...

.070541352850566718528... 
e 1

2
k 1
e
x
sin x dx
0
1
 3 
 log 3  log cosh

3 
2 

 (3k )  1
k 1
3k

 
1 
1

log1  3 

3 k 2  k 
 /4
1 
1
 1 
192 2 G   ( 3)     (3)       x 2 log tan x dx
0
3072 
4
 4 
1 .07061055633093042768... 
1 .07065009697711308205... 
ee
ee  1


1
1

 1
k 2k


 1
4 Li2   
 4
.0706857962810410866...

  cosh



H (2) k
1

k
k
2  3
k 1 4
k  0 4 ( k  1)
4   ( 2 )   ( 3)   ( 4 ) 

k
k 2

=
1

4
k 1 k ( k  1)
=
1
log(1  x) log3 x dx
6 0
5
1
 k4
1
3 2
9
3
 1 9  2
 1  log log 3  3(log 3  log 2)  4 Li2     Li2   
4
2
2
 2 2  3

(1) k 1 H k
=  k
k 1 2 k ( k  1)( k  2)
.070698031526694030437... 

 

3 2  log 2 2
log5 x dx
179 .0707322923047554885...    log 2
 
GR 4.264.3
18
( x  2)( x  1)
0
2
.07073553026306459368... 
9

sin 3k
 3
i
3i
 3i
.0707963267948966192... 

log1  e   log1  e   

2
2
k
k 1

1
sin k
1 .0707963267948966192... 

GR 1.441.1, GR 1.448.1
 
2
k
k 1

sin 1
sin 2 k
= arctan
  2
1  cos1
k
k 1
k

 1
2
 
2 .0707963267948966192... 
2
k  0  2 k  2


 k 
2
2


1  (2)  4 
( 2)  3  
.07084242088030439606... 
        
 10  
 5
1000 


1
log 2 x
x 4  x 1
( 1)
3 3
2
 log   k 2
8 4
3
k  2 2 ( k  1)

arctan 2 log 5 2
log x sin2 x
.07101537321990997509... 


 
dx
5
20
5
ex
0

k
.07090116891887671352... 

.07103320989006310636... 
 ( 1)
k 2
k
Fk 1  (2 k )  1 
k 2  2 k  1
3 .0710678118654752440...  5 2  4   k 

k 
k 0 8 
7 .0710678118654752440... 
50  5 2

326
( 1) k
 

.07130217810980315986...  120 
e
k  0 k !( k  6)

.071308742298512508874... 
 (3)
4
 8 log 2  2 log 2 
  
2

2
6
 6    log 2 (1  x ) log x dx
0

i
sin k
I 0 2 ei  I 0 2 e  i  
2
2
k  1 ( k !)

60
F
  (1) k 1 k 2kk
.07134363852556480380... 
841
4
k 1

1
 1
.0713495408493620774... 

  
16 8
k 1 ( 4 k  3)( 4 k  1)( 4 k  1)
1
log x
dx
=  2
2
2
  4 log x x  1
0
1 .07133434403336294393... 
1
J239
GR 4.282.8
.0714285714285714825
=
.0717737886118051393...

1
14

3
x2
  3x
dx
432
e  e3x
0
1
1 
1  
1 
    1 
 
  1 
2
2 
3 
3  

 2k 
1
.07179676972449082589...  7  4 3   k
 
k  1 16 ( k  1)  k 
.071791629712009706036... 
.07189823963991674726... 
k 2
1

k
3

1  ( 3)  3 
( 3)  7  
        
 8
 8
4096 
10  3

.07190275490382753558... 
8

 3k
log 3 x
1 x 4  1
 /4
 sin
2
x tan 2 x dx
0
 1 1
1
4
log    Lik     
4
3
 4 4
k 1 

log k
.071995909163798022179...   4
k 2 k  1

9
k2
.072000000000000000000 =
  ( 1) k 1 k
25
9
k 1

.07192051811294523186... 

.07202849079249295866... 
  ( k )  1
6

k 2
.07215195811269160758... 


 

 
2  9
( 1) k


2
12
k  0 ( k  3)
2

k 1 cos k
=  ( 1)

k2
k 1

log x
=  3
dx
x ( x  1)
1
.07246703342411321824... 

 8


 
=
e
3x
0
1 .07246703342411321824... 
.072700105960963691568... 

( 1) k


4
2
k 2 k  k

x
dx
 e2 x
2  3
12

GR 3.411.11
(1  x 2 ) log(1  x )
dx 
0
x
1

3



8 6 3
.07270478199838776757...  1  2 arctan


 ( 2k )  1
k 1
1

2
9

k
4
k 1


 9k
k 2
k 1
( 1)

( 2 k  1)
k
1
2
1


 9k
k 1
2
1
 18k  8
.07277708725845669284... 
1  ( 3)  3 
( 3)  4  
        
 5 
 10 
1000 


1
log 3 x
x 4  x 1
1
log k 
lim  log 2 n  

n   2
k 
k 1
n
.072815845483677...

.07284924190995026164... 
=
Berndt 8.17.2

 1  1
1
1
  1   
  2 2
      (  1) k  2 k   k

1 
1 
1
 (k )  1

  2    

 

k
k 2

.072900620536574583049... 
x log x
 1
1  Li2      
dx 
x

3
 3
0
.07296266134931404078... 
1

1

 5  3
    1   
2  4  4
 2k  (1) k 1
  k

k 1  k  4 ( 4k  1)

2

 2k  1
1
K      
1 .07318200714936437505... 
  4  k  0  k  64k
1
 1 
1 
 (1  i )   (1  i )     i      i   
.07320598580844476003... 
4
2 
 2 
2


 
.0732233047033631189...
cos2 x
dx
x
1
0
=
e

2 2

8
.07325515198082261749... 
.073262555554936721175... 
=


1 .07330357886990841153... 
 

 
=
755 .07330694419801687855... 
.07344910388202908912... 
cos  x
e 
2
0
x
1
dx
1  ( 3)  1 
( 3)  3  
        
 4
 4
20736 
4

e4


(1)
k 1
Prud. 2.5.38.9


1
log 3 x
x4  x2
k
4 k
k!
1 
 (k )  1  1

  2 1F1 1, 2, 3,   
(1) k

k
2 k
2k 

k 2
k  2 8k
1
9 
  1 log 
2
16 
120 I 6 ( 2 )  2670 I 7 ( 2 )  8520 I 8 ( 2 )  8000 I 9 ( 2 )  
3025 I10 ( 2 )  511I11 ( 2 )  38 I12 ( 2 )  I13 ( 2 ) 

k4


2
k 1 ( k !) ( k  5)
k 1

.07329070292368558994... 

7
4
2
cosh 2 
.07345979246907078119... 
2
18

2
12 3



1
  (12k  7)
k 1

(k!) 2  (2k )

(2k )!
k 1
3

log 2
.07355072789142418039... 



3
10
3 3

(1) k 1 H k
.073553956728532...
 

2
k 1 ( 2k  1)
2


1

2 
(12k  5) 
J346

1 .07351706431139209227... 
.07363107781851077903... 
.07366791204642548599... 
.07387736114946630558... 
.07388002973920462501... 
.07394180231599241053... 
.07399980675447242740... 
1 .07399980675447242740... 
.07400348967151741917... 
.07402868138609661224... 

 2k 
3
1
  k
 
128
k  0 4 ( 2 k  1)( 2 k  3)( 2 k  5)  k 
ci(  )

5
4
( 1) k 1

 
k
2
e
k 1 ( k  2 )! 2
35
48 2
 /4
8
11

  sin x tan 4 x dx
3 3 2
0
2
 k2 

 2


sinh 2 
k  1  k  1

1 .0741508456720383647...

1 .07418410593764418873... 
2
log 2 (1  x )
x ( x  1)( x  2)
2

27

=
2
log 2 2 log 3 2 log 3 2
 1
 1  5 (3)

 (log 2) Li2     Li3    
 2
 2
2
3
8
0
.074074074074074074074 =

CFG B5
   (1) (i )   (1) ( i )
sinh 2 

Hk
 2
 2
3  6 log   3 log 2     k
 3
 3
k  1 3 ( k  1)( k  2)
1
1
=
( 1) k

k  0 3k  8


k2
( 1) k 

2
k 0
k

k3
( 1) k  Li 2 ( 2)  Li 3 ( 2)

2
k 0
k

log 2 x
log 3 x
dx

1 x 4
1 x 4 dx

 2 (5)
2 ( k )
  5
 (10)
k 1 k
2 log 5
1
1  1/ 5


log
5
5
1  1/ 5
Titchmarsh 1.2.8

Fk Fk 3
k
k 1 4 k

( 1) k

3  ( k )  1 
k 2 k

.07418593219600418181... 
2 .07422504479637891391... 

  (1)

5 .074304749728368200293... 
1/ 5


1
1
 Li   k   k
3
k 2
1

2/5
 (1)   (1)3 / 5  (1) 4 / 5





k 1
x
 ( x) dx
1
.07438118377140325192... 
1
5
log 
3
4

1


k
k 1 3k 5

dx
 (3x  1)(3x  2)
1

1
14
7
( 1) k

csch
  2
2
7
2
k 0 2 k  7
3
log 2 7 (3)
1 .07442638721608040188... 

  (2) log 2 
3
4


1
H ( 3) k
1
 
= 2 Li3    2 k 3  

k
 2
k 1 2
k 1 2 k
.07442417922462205220... 



 
log 2 x
= 
dx 
2x2  x
1
1 .07446819773248816683... 

x2
0 2e x  1 dx
 e2 log1  e  2  

1
 ( k  1)e
k 0
k 1

1 csch1
(1)

  2 2

2
2
k 1 k   1
184
4
7 (3)
.07473988678897301721... 
   log


27
9
2
 4
1 .0747946000082483594...  Li2  
 5
2k

.07454093588033922743... 
1 .07483307215669442120... 
2
16

3 .07484406769435131887... 
2

1 (1)  1 
G
   

 4
2
16
16
27 3

k (k  1)
k 1  2k 
 
k 
(1) k 1 k 3
 (k  1)  1

k
k 1 2 ( k  1)


1

2 
k  0 ( 4 k  1)
1
log x dx

x4  1
0




1
(1  k 4 ) 2
k 1
k 2
 3  i 3 
2 1 1   3  i 3 
 

.074859404114557591023... 
   
 2  
3
6 3   2 



2 1 1
1
 
=

   (1)1 / 3    (1) 2 / 3   4
3
6 3
k 2 k  k
.074850088435720511891... 


 ( 1) k 1 k ((4 k )  1) 
 

k

1
5 

  1  k
4



 
=

 
=
1
1
k 2 k  k
2

5
 ( 1)  (3k  1)  1 
k 1
k 1
.074877594399054501993... 
35  3 2

72

1

2 
k  1 k ( k  4)
1
x
0
3
log(1  x ) log x dx 
.07500000000000000000
=
1 .07504760349992023872... 
.075075075075075075075 =
2 .07510976937815191446... 
.07512855644747464284... 
3
40

1
 K1 / 2  
e
 2
5
 B5
66

1

4
k  0 ( k !!)
1
 (3)
log(1  x 4 ) log x
dx
 
16
x
0

1 .07521916938666853671... 
 2(5)   d 2 (k )k  5
Titchmarsh 1.2.2
k 1

1
1
1
1 
( 1) k 1 k 2
cos

cos




k
4
2
2
2
k  1 ( 2 k  1)! 2
1 .07548231525329211758...  5 ( 3)  3 ( 2 )
( 4)
.0757451952714038687...   ( 2 )   ( 3) 
  ( 2 ) ( 3)  2 (5) 
4

( 1) k 1 k 3
4 log 3
.07580375925340625411...  1 
 2

3
2
k 1 27 k  3k

1
= 
2

9k
k 1 k  3k
2 

 3 
2

 2
.07522047803571148309... 
( 1) k 1 k 3

k !2 k
8 e
k 1

93
1
.07591836734693877551... 
 
1225
k 1 k ( k  2 )( k  7 )

( 1) k 1
31
1/ 4
.0760156614280973649...  20e

 

k
2
k 1 k ! 4 ( k  2 )
1

.07606159707840983298...  1 
90
.07581633246407917795... 
.07615901382553683827... 
1 .07615901382553683827... 

( 4)  1
( 4)

( 1) k 1 k ! 4 k
erfi 1  1  
e
( 2 k )!
k 1
1
k 1 k

dx

( 1) 4
erfi 1  
  x
e
 2k 
1 x
k 1
0 e
k ! 
k

.07621074481849448468... 

4


k 1

 (2 k  1)  1
2k  1


1
k 1
1
  2 log k  1  k  
k 2

Hk
 k ( k  1)
k 1
4
Berndt 2.5.4

 

log 2
= 1   

2

1
  arccoth k  k 
k 2

=
2  x e  x log x sin x dx
0
.07646332214806367238... 
i
i
1 

 3  i
 3  i
  

  1      1     
 2 
 2
 2 
4   2
(1) k
 

  1  log 2   3
k 2 k  k

37
( 1) k
.07648051389327864275...  log 2 
 b( 7)  

60
k 0 k  7

1
= 
2
k 1 4k  27 k  30



=
x
1
1 .07667404746858117413... 
=

8
GR 8.375
dx
 x7
 1

1

 2

2
2
e 1
2


1
1


( 1) k 1  (2 k )  1  Im (1  i )


2
2 k 1
k 1 k  1
coth  
J983

=
sin x dx
ex  1
0

sin log x 1
=
0 1  x dx

 1

1
2 .07667404746858117413... 
coth  

 2
2
2
2
e 1


1
3
 
=  2
   ( 1) k 1  (2 k )  1  Im (i )
1
2 k 1
k

k 0
1  5
 
.07668525568456209094... 
 5/ 2  2 
1

.07671320486001367265... 
1  log 2

4
 /2

 
=

0

1


2
k 1 16k  8k
sin 3 xlog(sin x)
1  sin 2 x
dx 

56
k2
  ( 1) k 1 k
729
8
k 1
1
.076923076923076923
=
13
2
.07697262446079313824...  G  G
.07681755829903978052... 

 2 ( 4k
k 1
k
1
2
 4k )

GR 4.386.2
.076993139764246844943... 

p prime


(k )
k 1
k

p4 
log (4 k )  
.077022725309366050752... 




 19  7 5 arctan 1  5   2 7  2 5 arc cot 1  5  3 3  5 arc coth 2  
 4 

20(3  5 ) 




k
1

(1) F k Fk 1
=  k
k 1 4 ( 2 k  1)( 2k  1)

1

.0770569031595942854...

 

 


=


 

9
  ( 3, 3) 
8
1
1 x 2 log 2 x
dx 
2 0 1  x
1
 ( k  3)
 ( 3) 
k 0
3

  (1, 3, 0)
GR 4.261.12
1 1 1

x2 y2 z 2
0 0 0 1  xyz dx dy dz

1
7
1
1
.07707533991362915215... 

log  arctanh
  2 k 1
( 2 k  1)
2
6
13
k  0 13

2
( 2 k  1) 2
.07710628438351061421... 
 
4
128
k  0 ( 4 k  2)

139
1
.077200000000000000000 =

 
1800
k 1 k ( k  3)( k  5)
=
 .07720043958926015531... 
.07721566490153286061... 
4
16
 

1 ( 3)  1 
   
256
 4
1
1

2

1
  log
2
0
x



log3 x
1 x 4 1 dx
x 
 dx
(1  x ) 2 
GR 4.283.4

x dx
(1  x )(e2 x  1)
0
=
2

1    ( x  1) dx
Andrews p. 87
2
1
.0772540493193693949...
0
.07730880240793844225... 
.077403064078615306111... 

4
 T( 4)
1260
9
 ( 3)
   log 2 

16
2

k2
 (2k  1)  1

k 2 k  1
Adamchik-Srivastava 2.24
.07743798717441510249... 
1   erf
1

2

( 1)
k
(2 k  1)
 k !4
k 1
k 1

1 .07746324137541851817... 
1
 k !! k
k 1
.07752071017393104727... 
3
log( 2 cos1) 

 ( 1)
k 1
.07755858036092111976... 
=
5 .07770625192980658253... 
.07773398731743422062... 
3 .07775629309491924963... 
.07777777777777777777
=
k 1
cos 2 k
k
2
2
1
log 5 
log

1

4
10
3 5
5 4 5


 (k )
1


k
2
k 2 5
k 1 25k  5k
9  1/ 2

(1) k 1
5
1
 log  2  
k
4
2
k 1 k ( 2k  1) 4

 
0 ( k )2
 0 ( mn )




k
2
2 mn
k 1
n 1 m 1
4 arctan
7
90
( 1) k 1 e k

k!
k 0


.07784610394702787502... 
 cos e 
.07792440345582405854... 
2  2 cos1  sin 1   ( 1) k

k 1
1 .07792813674185519486... 
=
=
GR 1.441.4
105
 1  p  

4
p prime

( k )
( 4)
  4
k
(8)
k 1
q
k
( k  1)( 2 k  1)!
4
4
Berndt 5.28
Titchmarsh 1.2.7

q squarefree
.07795867267256012493... 
8
2
 2 log 
9
3
( 1) k 1 k 2
k
( k  1)
k 0

2
2

50G  43 1
1  (2 k  1)!!
.07796606266976290757... 
  

 
128
7
k  1 2 k  7  ( 2 k )!! 
1
2 .0780869212350275376... 
log 


1
 (k )

1 .07818872957581848276...  
   e1/ k  1  

k
k 2 k !
k 1
.07820534411412707043... 


2e
3

cos 3x
dx
2
1
0
x
J385

( 1) k 1
1
2
2
 csch sech  sinh  2   2    2 2
2
12
2
2
k 1 k ( k   )
i
i


1 i 
1 i 
1 .07844201083203592720...   1     1     
  

 2
 2
 2 
 2 

3

( 1) k 1 k
.07844984105855446265... 
   k
4 3 8
k 1 3 ( 2 k  1)
.0782055828604531093...

  sin  2  sinh  2 
.07847757966713683832... 

2 2 cosh  2  cos 2

1

1

( 1) k 1  (4 k )  1

4
1

k 2
k 1

17819
1
.0785670194003527336860 =

 
226800
k 1 k ( k  1)( k  10)

2

 (2 k  1)  1
.078679152573139970497... 
 log 2   6 (1)  

3
2
k 2
k 1

1
1
 3 
=    k log 1  2   k 


k 
2k 
k 2 


 
=
k
1
.07882641859684234467... 
ci (1)    log 2  cos1 log 2 
 log 2 x sin x dx
0

.07889835002124918101... 

k
k

 1  2
k 1


 (2k  1)  1
2
 log 2   6 (1)  
k2
3
2
k 1

1
1

.07896210586005002361... 
4 2  24 log 2 2  27    k
2
12
k 1 2 ( k  2)

3 ( 3)  2
( 1) k 1 k
.07907564394558249581... 

 
3
4
12
k 1 ( k  1)
 1 

.079109873067335629765...  log  ( 4)   log
4 
p prime
1 p 
1 .07891552135194265398... 
. 07912958763598119159... 
HW Sec. 17.7




3 G  3
 3 2
3
 1  3
 3

(1) 3 / 4 (1)   
(1)1 / 4 (1)   
cot 
log 1  (1) 3 / 4 
512
32
512
8
64
 8  512
8







3
3 (3)
7 3
 5  3
(1)1 / 4 (1)   
(1) 3 / 4 (1)    Li3  (1) 3 / 4  

512
2
8 2
 8  512
 /8

 


0
.07921357989350166466... 
=
.07921666666666666666
=
.079221397565207165999... 
.07924038639496914327... 
x3
dx
sin 2 x

 (k )
  4 
log        ( 1) k k 
  3  3
3 k
k 2

3k  1 
 1
  log


3k 
k 1  3k
7

96

Dingle 3.37
1  dx
2 9
x

 log1  x
1
53 (6) 3 2 (3)

24
2

1
1
1 
( 1) k k
 2 sin
 cos
  
k 
2
2
2
k  1 ( 2 k  1)! 2
MHS (3,1,2) 


.079332033399540433998... 
  (3x)  1 dx
1
( 1) k k


k 1 ( k  1)( k  2 )


1
 ( 2 k  1)
= 


3
16 k
k 1 16 k  k
k 1
.07944154167983592825... 

 
3 log 2  2 

(1) k 1

k 1  2k  k
  2 k (2k  1)
k 

=
.07957747154594766788... 
1

4


cos( / 2 x)
dx
2 x
e

1
0
1 ( 2 )  3
.07970265229714766528...  
  
 5
125

3 log 3
1
 4
.07998284307169517701...  3 

  (1)   
 3
2
3
2 3
1
=
 log(1  x ) log x dx
3
0
.080000000000000000000 =
2

25

 (k )
 10
k 1
k
1

.080039732245114496725... 
log 2 x
251
   ( 2) (4)   5
dx
2 (3) 
x  x4
108
1

1
 k (3k  1)
k 1
2
1
=
x 3 log x
dx
1 x

0
32 / 3  3 9
7
1 .08012344973464337183... 
 1
6
2 .08008382305190411453... 
.08017209138601023641... 
.0803395836283518706...

4
1215
k 1


1

 (k )  1
k
3
k 2
.08035760321666974058... 
( 2 k  1)!!
 (2 k )!!7
k

log 3 x
dx
x4  x
log 3

1

 
2
6 3 6

=



1
 (3k  2)(3k  3)
k 1

1
 

k  2 3k ( 3k  1)
log 2 2   log 2 


(1)
k 1

x
1
6
dx
 x5  x 4
log k
k
k 2



4
(1) k 1


 log
   k  (4k )  1 
k 1
 cosh  2  cos  2 

1

log 1  4 

 k 
k 2


1
k 1
 (5k  1)  1

4
 1   ( 1)
k 2 k  k
k 1

1
H
96 log 2  18 log 2 2  55   ( 1) k 1 k

36
k 4
k 1

 
1 ( k )
 ( mn )
   mn

k
3
n 1 m  1 2
k 1
3

log k

k  2 6 k ( k  1)
1
2 cos1   ei  e  i  
2

1
1
k B2 k




2
4 2( e  1)
k  1 ( 2 k )!
2
.080362945260742636423... 

 
=
.08037423860937130786... 
.080388196756309001668... 
.0805396774868055356...

1 .08056937041937356833... 
1 .08060461173627943480... 
.08065155633076705272... 
.080836672802165433362... 

10


1  x2
50  10 5  5   2 arctan( x 5 ) dx
x
0
1
( 1) k

4
k  2 k log k

.081045586232111422...

.08106146679532725822... 
1  log 2 

 1
1
  k  1  2k   ( k )  1 
k 2
K ex. 125
2
=
  log  ( x ) dx 
GR 6.441
1
 1
1
1  dx

 

 
 log x 1  x 2  log x
0
1
=
.081019302162163287639... 
GR 4.283.2
3
1  1
1
log   k
2
5 k 1 3 k
5
log5 x
8 6
( 5)
dx
 120 (6)   1   
63
1 x
0
1
122 .081167438133896765742... 
.08126850326921835705... 
13 


15 4
( 1) k


k 0 2k  7



1
dx

8
x  x6
 /4
 tan
6
x dx
0

.081317489046618966975... 
  (2k  1)  1 dx
1



.08134370606024783679... 
i
6
.08150589160708211343... 
log 3
1
 log 2 

2
4
e
  3 (3)
k

 
1 coth 



 cot
3  4  i   cot
3  4  i    
2
6

2

 2
2


k
=  6
  ( 1) k 1  (6k  2)  1
k 2 k  1
k 1

1
( 2 k )!!
.08135442427619763926...  9 2 arcsin  1  
k
3
k 1 ( 2 k  1)!! 9 ( 2 k  1)

1
x dx
2 .08136898100560779787... 

2

log 2
2x
0
3
1 .08163898093058134474... 
.081812308687234198918... 
32
5

192


0
1

 ( 1)
k 0


x
dx
(e  1)(e x  2)
x

1
1


3 
3
 (2 k  1)
( 2 k  2) 
(sin x  x cos x)3
dx
x7

k3

(1) k 1  (7 k  3)  1


7
k 2 k  1
k 1
(1) k
4
1 
1 .08194757986905618266... 
sin  
2 k  0 k!(k  12 )16 k


3
k
.08197662162246572967...  15 log  6   k
2
k 1 3 ( k  1)( k  2 )
1
1
3e
1
1


.081976706869326424385... 
 coth  1 
2
2(e  1)
2
e 1
2
Prud. 2.5.29.24

.08183283282780650656... 

Bk
 (2k )!
k 1
1 .081976706869326424385... 
=
.082031250000000000000 =
.08217557586763947333... 
1
1
e1/ 2  e 1/ 2
coth 
2
2
2 e1/ 2  e 1/ 2 

B2 k

k  0 ( 2 k )!

21
k2
  ( 1) k 1 k
256
7
k 1

1

log 2
( 1) k


 

2 6 3
6
k  0 6k  8
.0822405264650125059...

1 .08232323371113819152... 

x
9
1
dx
 x3
log 2
log 3
 1
 1
 log 2 log 3 
  Li2    Li2   
 3
 2
2
2
2
.08220097694658271483... 
AS 4.5.67
2
12

4
90
2


log 2 2 1
1
  k

2
2
2 k 1 2 (2k  4k  2)

2
k 2
1
k2
k
  (4)  MHS (2,1,1)

=
H ( 3) k

k 1 k ( k  1)
=
36
17
=
1
k 1 4  2 k 
k  
k 

1
9
3
k 1 4  2k 
k  
k 


42 2 
1 .082392200292393968799... 
.0824593807002189801...

=
.08264462809917355372... 

1 k 1 1

 2
k 1 2  2k  j 1 j
k  
k 

( 1) k 

1



4k  6
k 0 


7
2
4 
.08248290463863016366...  5
3
.08257027796642312563... 
Borwein-Devlin, p. 42
( 1) k k 2  2 k  1

 

8k  k 
k 0

( 1) k k  2 k 
 

k
k  0 8 ( k  1)  k 

1 
 2 sin  2  sinh  2
 coth  
16 8
8 cos  2  cosh  2



(
8
k

4
)

1


  k 4 1 k  4

k 1
k 2
k 1

10
( 1) k
 
121
10k
k 1

(1) k 1 k
 (k  1)  1
.082681391910818588188...  4  4 log 2  log   
k 1 ( k  1)( k  2)

1 .08269110766346817971... 
1 Hk
 k! 
2

H ek


k 1 k!

.082710571850225464607... 
1 .082762193260924580122... 
6 .082762530298219680...
x log x
1 
6

dx 
   log 2  1  2  (2)     2x
e
24 

1


0
6e


2
 lim
n 
1
log n


( 6k  2)  1
=
=
.082936658236923116009... 

 
=
15 .08305708802913518542... 
122 .08307674455303501887... 
.0831028751794280558...

.08320000000000000000
=
.08326043967407472341... 
1
 k 3
   k 2 log(1  k 6 ) 
1 
1
1
1
1 
cosh 1 / 4 sin 1 / 4  cos 1 / 4 sinh 1 / 4  
1/ 4 
2 
2
2
2
16  2 
1
1
1
sin 1 / 4 sinh 1 / 4 

2
2
8 2

2k k 2
(1) k 1

(4k )!
k 1
7
6
4


41
k k k
k 1 ( 2 k  1)
  ( 1)
  ( 1)
( k  1)!
( k  1)!
e
k 1
k 1
5


log k
   (5) ( m)

k  2 k ( k  1)
m 2

( 1) k 1 Hk
5
5
log 2   k
2
6
k 1 5 ( k  1)

276
k4
  ( 1) k k
3125
4
k 1
2
3
 log 2 log 2
 1  15

 2 Li3      (3)
 2  36
18
3
log 2 (1  x )
0 x( x  3) dx
1
=
k 2

4
k
k 2
1
7 (3)  3
 3
  ( 2)   

64
8
32
 4
7 (3) i
 Li3 (i )  Li3 (i ) 
8
2
1 2

2
x log x
log 2 x
dx

0 1  x 4
1 x 4  1 dx
k 1
.08285364400527561924... 
p prime  n
  (7k  3)  1   k
k 1

.08282567572904228772... 
1
37 

.08282126964669066634... 
 1  p 
Berndt 2.9.7

.083331464003213147...


 (k )
k
k 2
.083333333333333333333 =

=
4
1

12
e
1 .08334160052948288729... 
.08334985974162102786... 
1 .08334986772601479943... 
.08335608393974190903... 
2 .08338294068338349588... 
k

  (6k  2)  1
log  (4k )

k 1

k
k 2
4
x
1
 k 2
Prud. 2.5.38.9
1

 (4 k )  1


 1
4

k
k 1
k 2

1
cos1  cosh 1  2 
k  0 ( 4 k )!


.08339856386374852153... 
k 1

25
 H4   4 k  (2 k )  1  1 
12
k 1

k !!

2
k 1 ( k )!

257
k3

  ( 2)  9 ( 3)  27 ( 4)  27 (5)

5
32
k 1 ( k  3)

k!

2
k 1 ( k )!
1
5
log(1  x )
 2
 1
log log 2  Li2     Li2     
 3
 3
3
x4
0

.08337091074632371852... 
 (k )
dx
2
0
2 .083333333333333333333 =

  (4)
2

 Li  k
2
 1  2 
k 2
k 1
1 .08343255638471000236... 
.08357849190965680633... 
1 .08368031294113097595... 
.08371538929982973031... 
57 .08391839763994994257... 
8103 .083927575384007709997... 
1 .08395487733873059...


1
cos 4 2  cosh 4 2 
2

 (k )
1
  p

4
k 2 k
p prime 4  1
2k

k  0 ( 4 k )!

1
cosh  2  cos  2


1  4 

2
k 
4
k 2 
2

47 log 2 21 log 2 85
k 2 Hk


  k
2
2
4
k 1 2 ( k  1)( k  2 )( k  3)
21e
e9

H ( 3) 3/ 2

( 1) k
1 G  
2
k 1 ( 2 k  1)

.0840344058227809849...


k  ( 2 k  1)
16 k



k
2
2
16
k 1
k 1 (16 k  1)
1  k 
3

k   k  1, 
8 k 1 2 
4

=
=

=
log x
dx
4
 x2
x
1
.08403901165073792345... 
=
.084069508727655996461... 
3 .084251375340424568386... 
.08427394416468169797... 
B2 k
1 

 2 log 2 sin    (1)k 1
2  k 1
(2k )! k


cos k
2
k
k 1
1
5 2

16

8

.08439484441592982708... 
.084403571029688132...


2

32
.084410950559573886889... 
.08444796251617794106... 
=

k 1
2
.0848054232899400527...

2
( 4 k  1)
)  1


k
k 2
 e


k 4
4
1
 k 1

1
k 2

 cos1  si (1)   
log 2  log 2  i   log 2  i 

 (3k  1)  1
k
2
1
1

.084756139143774040366...  1  cot
2
2

1
 (4 k  3)(4 k  2)
k!
k 1
.0848049724711137773...
Borwein-Devlin, p. 82
cos x
dx 
x2
2
1


(4 k )  1
4
log
 
   log(1  k 4 )
k
sinh 
k 1
k 2

.08452940066786559145... 

 (4 k )  1
k 1




k4

2
3
k  1 ( k  1 / 4)
  (5k  1)  1
k 1

AS 4.3.71

  ( k
k 2

 1
kk



 2 k
k!e k
k 1 

1
  
2  2 
2

3

.08434693861468627076... 
Adamchik (29)
e

1
 1
3 

 k Li  k

k 2
2
( 1) k 1 B2 k

(2 k )!
k 1

 i log i  e log i
4 2
log 4 2
 1  21

log 2 2 
 6 Li4     (3) log 2 
 2
15
4
2
4
2
/4
2
J271
log 3 (1  x )
0 x( x  1) dx
17
.085000000000000000000 =
200
1

.08502158796965488786... 
2 arctan


2 4

k
4 .0851130076928927352...  
k  2 dk
1

 
=
2
1 .08514266435747008433... 
 /4

0
sin 2 x
dx
1  cos2 x
dx
 ( x)
1
.08515182106984328682... 
178
1
 
225 6
2
1
 (2 k  1)!!




(2 k )!  2 k  6
k 1 

2
J385
7 (3)
.085409916156959454015...  12 
 4 log 2 
   log(1  x 2 ) log 2 x dx
2
2
0

41
1
.08541600000000000000 =
 
480
k 1 k ( k  2 )( k  6)

2
5
log( x 2  1)
.08542507002951019287...   
1   3
dx
24
x ( x  1)
0
1
5 2
1 .08542507002951019287...   

24
1 .08544164127260700187... 
=

log( x 2  1)
0 x 2 ( x  1) dx 

1
1
 

k
2
k  0 ( 2 k  1)! 2

1 

1  2 2 

2 k 
k 1 
2 sinh
1 
 11  
5
1 .085495292598484041001... 
16G  8 2 3   (1)     (1)    
144 
 12  
 12 

.0855033697020790202...

 4 (4
k 1
k
2
k 1
k2
1
log x dx
 1  x 2
2
 1)k

1 .08558178301061517156... 
x
1
k
5 1
    Ei(1) 
4 3
3
20 .08553692318766774093...  e

1
2 .08556188259741465263...    ( k )
k 1 2

2k
.0855209595315045441...

( 2 k  1)
(1) k 1

2
k 1 k!( k  2)

Berndt Sec. 4.6

u2 ( k )
k
k 1 2

log k
.085590472112095541833...   4
2 
k 2 k  k
=
.085628268201073866915... 
.08564884826472105334... 
.08569886194979188645... 

 


 
=

 (3)
2

2
24

5

48

HkHk
 k (k  1)(k  2)(k  3)
log 2

3

 
3
3 3 4
k 1
( 1) k


k  0 3k  7


x
1
8
dx
 x5
 5  i 3
log 2 1 1  i 3   3  i 3 
 
 
 
 
3
6
12   4 
 4  


 5  i 3
1 i 3   3  i 3
 
 
 
12   4 
 4  
(1) k

3
k 2 k  1

( 1) k

k
k 2 2 ( k )

.08580466432218341253... 
1 .08586087978647216963... 
.08597257226217569949... 
.08612854633416616715... 
1 .08616126963048755696... 
3 .08616126963048755696... 

 3
  
 4
2

1
sinh 1
cos k

 1   k

2  cosh 1  cos1 
k 1 e
  3 log 2 
3
( 1) k 1
  3
360
k 1 k sinh  k

1
1
e   2  2 cosh 1  2  
e
k  0 ( 2 k  1)!( k  1)

1
1
1
e   2 cosh 1  4 sinh 2  2 
e
2
k  0 ( 2k )!

1  sin 1
1
.086213731989392372877... 
csc2 
8
2
3

( 1) k 1 k B2 k

(2 k )!
k 1

log 2 log 2 2
.08621540796174563565... 
 


48 8
4
4
.086266738334054414697...  sech 
1 .086405770512168056865... 
.086413725487291025098... 
GR 8.366.5
 /2
 log sin x  sin
2
2
x dx
0
7( 3)
4 2
(1  log 2)  2 log 2 


4
64
4
 /4
 log 2 G
cos x  sin x

 
x dx
4
2
cos x  sin x
0

Hk
 (2 k  1)
k 1
3
 /4
=

0
x2
dx
(cos x  sin x) 2
 /4
=
 log cos x dx

GR 4.224.5
0
.086419753086419753
=
7

81

.086642070528904716459... 

.086662976265709412933... 
.086668878030070078995... 
1 .086766477496790350386... 
.086805555555555555555 =
.08683457121199725450... 

.08685154725406980213... 
.086863488525199606126... 
1 .08688690244223048609... 
.087011745431634613805... 
2 .087065228634532959845... 
9
k 1
 ( x)  1
2
.086643397569993163677... 
 (k )

2x
1
k
dx


log 2
1
1
  k 3  
2
8
k
k 1 2
k 1 32k  16k


7 
1
 coth    4
  (4 k )  1
8 4
k 2 k  1
k 1


 (3k  1)  1
1 3
  ek  1

k!
k 1
k 2 k
2

 2k 

1
2
8
 8 log 2  4 log 2     k
3
3
k  0  k  4 ( k  1)



k3

k 1 ( k  1)( k  2 )( k  3)( k  4 )( k  5)

1

k  0 k ( k  1)!( k  2 )!( k  3)!
25

288
1  2 log 2



8 32
2

 (3k  1)  1
k 1
k


log x
 ( x  1) ( x  1) dx 
log1  k 
 
k
3
1
k 2
 1
3
3
 
log  
csc 
2
2
 2

64
3

 3   (2 k )  1

 

 
k
k 1 2

k
 /2
( 2 log 2  1)  
e2/e 
 log(sin x) sin
2
x cos2 x dx
0

2
k
 k !e
k 0
k

2arcsinh1 2
( 1) k 1 k  2 k 
  k
 
4
k  1 4 ( 2 k  1)  k 

1 
1
 3 log 2
( k )
.08744053288131686313...  1 

  2
  ( 1) k k 
8
4
4 k 1 4 k  k
4
k 2
.08713340291649775042... 


 
=
x
5
1
.087463556851311953353... 
dx
 x  x3  x2
4
30

343

 ( 1) k 1
k 1
k2
6k

11
6k  8
1 .087473569826260952066...   ( 3) 
  5
4
3
96
k 1 k  6 k  8 k

1
1
1 
k
.08757509762437455880... 
 2 sinh
 cosh
  
k 
2
2
2
k  1 ( 2 k  1)! 2

1 .087750469214439700994... 
.087764352700090443963... 
.08777477515499598892... 
.087776475955337266058... 
.087811230536858587545... 
2 .087811230536858587545... 
.087824916296374931495... 
.087836323856249096291... 
.087981169118280642883... 
1 .088000000000000000000 =
.08800306461253316452... 
13 8
13

 (8) 
113400
12
H ( 4) k

k4
k 1



1
1
( 4 k )





k
k
k
4k
k 2 4  1
k 1 4 ( 4  1)
k 2

 ( 2k )
log 
1 3 (3)




2
k
3
18
2
k 1 4 k ( 2k  3)
 log G

k
2 ( 2 )   ( 3)  2  
3
k 1 ( k  1) ( k  2 )


( k  1) ( k )
3k  1
2 ( 2 )   ( 3)   3


2
2 k 2
k 1 k ( 2 k  1)
k 4

1
1
2  2 7 arcsin
 
2 2
k 1  2 k  k
  2 (2 k  1) k
k
24 

( 1) k


k  0 k !( k  5)


log 2 2

 
6
3
3

136

125




3
pf (k )
k4
J1 2 3
k 1
.08802954295211401722... 
65

e


.088011138960...

0
2

e
log 4 x
1 x 2 dx
1  dx
6 4 
x
 log1  x
1
2
x sin x
dx
ex


 ( 1) k
k 0
1
 7
 cos

6
2


3k
k !( k  1)!
7
 1  (2k  1)
k 1
2



.08806818962515232728... 
6 .08806818962515232728... 

4
16


6 
1

4
log 3 x
dx
x4  x2
1
3
log 3 x dx
0 x 2  1 
log x dx
  4

x 1
16
0

=
x 3 dx
0 e x  e x
1 .08811621992853265180... 
4

sinh 

k4
 (4 k )  1




2
2
i
2
i




exp




4
k
k 2 k  1
k 1

.08825696421567695798... 
Y0 (1)
.08839538425779600797... 
23 log 2


72
3
.088424302496173583342... 
1  x2
0 log(1  x) (1  x)4 dx 
1
GR 4.291.23
  ( )  1 where S is the set of all non-trivial integer powers
 S

.0884395847555822353...

 ( k ) ( k )  1 
k 2

1

k  j 1 k j

.08848338245436871429... 
MHS (4,2) 
.088486758123283148089... 
3 log 6
19
 log    

6
2
.08854928745850453352... 
1
5cos1  3sin1 
2
.08856473461835375506... 
1

log 2 tan  

2
.08876648328794339088... 
1 2


2 24
4 2
(1) k 1 2k k 2
 (k  1)  1

k 1 ( k  1)( k  2)

( 1) k 1 k

k  1 ( 2 k )!( 2 k  3)


 ( 1)
k 1
k 1
(2 2 k 1  1) B2 k
(2 k )! k
AS 4.3.73
1
 x log(1  x
2
) log x dx
0
1
=
  x log x log(1  x ) dx
0
1 .08879304515180106525... 
 log 2
2


 2k 
  k  4
k 0
1
=
arcsin x
0 x dx 
1
=

0
x arcsin x
dx
1  x2
 /2
k
1
(2 k  1) 2
 log sin x dx
0
Berndt 9.16.3
1
=
log x

dx
1  x2
0
GR 4.241.7
1  x 2 dx
0 log x 1  x 2
1
=
GR 4.298.8
 /2
=
 log(2 tan x) dx
GR 4.227.3
0
 /4
=
 log(tan x  cot x) dx
GR 4.227.15
0

=
dx
 log(coth x) coth x
GR 4.375.2
0
.088888888888888888888 =
4
45
.08894755261372487720... 
 k ( k  1)    k  k
( k )


k 2
1
j
k 2 j 2

=

j
 1

  ( 1)  (mk  m)  1
k
k 1 m  2
.08904125208589587299... 
1
=

0
.08904862254808623221... 
.0890738558907803451...

2

1
x 2 dx
1 e3x  1
2
3 1
 
16 2

sin k cos3 k


k
k 1
1
   erf (1)  1 
e
 (k )  1
 ( k  1)
k 2
.089222988242575037903... 

log 2 x
dx 
x4  x
x log x
dx
1  x3

.089175637887570703178... 

2 (3)

27
2




sin k cos 4 k

k
k 1
e x
1 x 2 dx

2

 1
  k Li  k   1
2
k 2
2
1
 2  9 log 2 2  3 log 3  2 log 2  3 log 2 log 3  4 Li 2   
3
4
1

 
=
  log1  x / 2  log(1  x / 2) dx
0
1 .0892917406337479395...

e   1   1 
 I 0    I1    
2   2   2  

.08936964648404897444... 
 ( 1)
k 2
k
 (k )
k
2 k
2


k  2k 
  

k
k 1 k!4  k 



1
1
  Li   2k   2 k 
k 1
2

(2k  1)!!
 (2k )!! (k  1)!
k 1
1 .08942941322482232241... 
.08944271909999158786... 
2  5  1  3 
3 1  1 
 K
     
 
3
2 3  2
 4  4

1
( 1) k k  2 k 
 
 
k
k
5 5
k  0 16
2

3

1

1  x 4 dx 
0
  dx 
1
  e
 

 0 ( x) 
k  0  ( k  1)

.08948841356947412986... 
dx
0  ( x) 
.08953865022801158500... 
2 log 2
319


7
2940
( 1) k 1

2
k 1 k  7 k

1
2


 log(1  2 )   
3
4 
2


( 1) k
dx
= 
  8
x  x4
k 0 4k  7
1
.08958558613365280915... 

 
.089612468468018277509... 
1 .08970831204984968123... 
10
2
 8 log 
3
3
90
1


4
180  
2  ( 4)

.089738549735380789530... 
(1) k k


k
k  0 2 ( k  2)



k 1

10

45k
k 1 k  15k
3 

 15 
2
 2

f (k )

Titchmarsh 1.2.15
k4

1
 k (k  1)(k  2) log k 
k 2

1 .08979902432874075413... 

 (k )  1
k 2
7 .0898154036220641092...

1 .08997420836724444733... 
k
3


 1
1
 Li  k   k
k 2
3
4 
 1
 erfi   
 2

 k !4
k 0
k
1

(2 k  1)
2

9

100
.0900000000000000000000 =
 ( 1)
k
9k
k
k 1
 (k )


2
1

   log 1  

( k  1)
2k 
k 2
k 1 k
1 log 2 1 
 1 
 1 
.090029402624249725308... 

   i      i       i       i   
 2 
 2 
2
2
8
2
2 .09002880877233363966... 
k 2


 
sin 2 x
0 e x e x  1) dx
=
 1
.09005466571394460992...   ( 2)  log 2  2 Li3   
 2
2


2
k 1
k
k
( k  1) 3
57
log x
 
dx
4
1

x
512 2
0
2
1
 5
.09016994374947424102... 

11  5 5

2
5 5 1
3 .09016994374947424102... 
 10 cos
 10 cos  5 
5
5
2
5
4
24 .09013647349572026364... 


11  5 5
 5
2

47
(1) k
.09018615277338802392... 
 log 2  

60
k 0 k  6

1  2k  1


k
 k 
5
k 0
11 .09016994374947424102... 


 
x
=
7
1
k 1
2
1

 18k  20
dx
 x6
13
.090277777777777777777 =

144


 4k

1
 k ( k  3)( k  4) 
k 1
.090308644735282059644... 
 




 


1 5 
10 1  5 arc cot 2  2 5  2 5 arc coth 1  5   5  5 arctan h

4 
20 1  5 

Fk Fk
=  k
k 1 4 ( 2 k  1)( 2k  1)

1
1 .09033141072736823003... 
coth

2


k2


k
k  2 2 ( k  2)

 2k  (1) k 1
1 1 4

 .09039575736110422609...  1  2 F1  , , ,  1     k
3 2 3
 k 1  k  4 (3k  1)
1 .090354888959124950676...  16 log 2  10 
3

  
.090397308105830379520... 
 7
343
1
( 1, 0 ) 
.090411664984819155127...   Li
 
0 
 2
.090430601039857489999... 
3
(1) k
log k

k
k 1 2

1 (1)  2  1
1
    
2 
 3 4
9
k  2 ( 3k  1)

( k  1) ( k )  1
3k
k 2
1
4
.09045749511556154465...  2  4 arctanh  log

2
3


 
=


4
k 1
k
1
k ( 2 k  1)

.09049312475579154852... 
 csch( k )
k 1
.09060992991398834735... 
 
1 
 
 log
2
 2 

 (k )  1
k 2
2k k

=
1 1
 
   log 1     
 2k  2k 
k 2 

1

e 2 I 2 (2)  1F1  , 2, 4  
2




 
5 .090678729317165623...
.09079510421563956943... 
1
1
 log(e  1) 
e
.0909090909090909090909 =
1 .0909090909090909090909 =
.09094568176679733473... 
.09095037993576241381... 
.091160778396977313106... 
1 .091230258953122953094... 

ck
 k!
k 1
dx

x
 1)
 e (e
x
1
k 1
e(e  1)
( 1) k 2


(e  1) 3
ek
k 1
1
11
12  F3k

11 k 1 6 k
2
7

 2  7 ( 3)
k
 
3
16
k 1 ( 2 k  1)

1
6
1
1

log  arctanh
  2 k 1
2
5
11
( 2 k  1)
k  0 11

.090857747672948409442... 

1  2k 
  

k  0 ( k  1)!  k 


tan
 21 1



k
2
1
 3k  3
2
5 k 0
21
1
.09128249034461018451...  
15  4 2  12 coth   12 2csc h 2 
192

K148
  k  (4k )  1 

k 1


 
=
k
k 2
2
3
1
(1  k  4 ) 2

(1) k
1


.091385225936012579804... 
6


2
108 k  0 (3k  3)
k 1  2k  2
  k
k 
.09140914229522617680... 
5e 


log x

4
x
x
1

k2


k 1 ( k  3)!
27

2

1 .091537956897273237144... 
1 5 5 1
sin x 5
    5 dx
8
2
5 0 x
3 log 2  7  2  8G  k ( (k )  1)
  

.09158433725626578896... 
4
8 36
16
4k
k 2

4 cos1  1
cos k
.09159548440788735838... 
  k
17  8 cos1
k 1 4

1
.09160199373136531664...   4
k 2 k  2
1
G
1
 7
 3

2 .091772325701657557483... 
sech
cosh
2
2
2
1 .09174406370390610145... 
.09178052520743498628... 
.09180726255210907564... 
coth 
1


4
2
 
=
k
k 2
0
1

4
k

  (3k  1)  1
k 1
.09197896550006800954... 
1 (1)  4 
   
 5
25
7 .09215686274509803
=
k 1
sin 2 x
x
x
 1
.09189265634688859583... 
.092000000000000000000 =

 1  k
 2e e
1
1
1 
 2 cosh
 3 2 sinh
 
8
2
2
.09199966835037523246... 

 3  i 3
 3  i 3
1
 2  2   
 
 
3
 2 
 2  



k2  k  2


2
k 1 k  k  1

1

 1
23

250
2


1

2 
k  1 (5k  1)
cos  x
ex
0

3617
 B8 
510

k2

k
k  1 ( 2 k  1)! 2
x
1

dx
1
5
2
1


 k  1

 .09216017129775954758... 
5
24 

64
5
.0922121261498864638...
.0922509828375030002...



4
.09246968585321224373... 
.0925925925925925925
=
.09259647551900505426... 

1  dx
2 7 
x
 log1  x
1

log 4 x
x4  x2

1

 ( 1)
 log 2 
k
k 0
log(1  x )
0 (1  x)2 dx 
1
=

log 2
5


3
36
2

k

( k  1)( 2 k  1)

1  dx
2 
2
 ( x  1)
 log1  x
1

log 3  3 1
1

  
4
36
3
k 1 ( 3k  1)( 3k  3)
2

5
k
  ( 1) k 1 k 
54
5
k 1

(2 k )k

2k
k 1

1 .092598427320056703295... 
 (3k  1)
k 1
.09262900490322795243... 
4  8 log 2  17

12
(1) k 1


k 1 k ( 2k  1)( k  2)


.092641694853720059006... 
.09266474976763326697... 
1 .09267147640207146772... 
1 .09268740912914940258... 
1 .09295817894065074460... 
15

12


(2 k  2)  1
log(1  k 2 )




k
k2
k 1
k 2

( k !) 2
8
1
arcsin
 
k
7
2 2
k  0 ( 2 k  1)! 2

1

3
k 2 k  7
  2 log 2  12 log Glaisher 
16  4

 1
.09296716692904050529...  6 Li3    2 
 3

.09309836571057463192... 

log 2 x
1 3x 3  x 2 dx

 e1/ k ( k  1) 1 
k 2 ( k )


 


k!
k2
k
k 2
k 1 
1


 log 10


cot 
 
2 10 20
5
10

6 .093047859391469...
2
 2k 
1
   

2
k
k  0  k  16 ( k  1)


 k  2  (2k  1)  1
k 1
k
=
=
1
2
 1
4
2
cos log sin  cos log sin
5
5
5 5
5
5

k

( 1)
dx
  8

x  x3
k  0 5k  7
1


3
(1) k  2k  1
I 2 (2)


 1F1 ( ,2,4)  
.0932390333047333804... 
2
e2
k!  k 
k 0

21
1
.09328182845904523536...  e 
 
8
k  0 k !( k  2 )( k  4 )

18G  13 1
 (2 k  1)!! 1
  
.09331639977950052414... 


32
5
(2 k )!  2 k  5
k 1 
.09334770577173107510... 
22
8 
e
1/ 3

 ( 1) k 1
k 1
J385
k3
k !( k  2)

3e
pf (k )
 k 
2
3
k 0

5  6 log 2
1
.09345743518225868261... 
 
9
k 1 k ( 2 k  1)( 2 k  3)


3
k2
(k )
.093750000000000000000 =
  ( 1) k 1 k  Li 2 ( 3)   k
32
3
k 1
k 1 8  1

1
 (k )  1
.093840726775329790918... 
log 32    5   ( 1) k

k ( k  1)
2
k 2
2 .09341863762913429294... 
2 k 1
1
2 
 ( 2k )
k 1 2
sin
(
1
)





k k 1
(2k  1)!
k 1 k

2 .09392390426793432431... 
11 4  3 5 2 4 5



 
360
4
6
2
6
4

k  1 cos k
  
=  ( 1)
k4
k 1

9 (3) log 2 1
 (2 k )



.094129189912233448980... 

2
k
4
2
6
k  1 4 ( 2 k  3)
.094033185418749513878... 


.09415865279831080583... 
1  2 log 2  log 2 2  (1  log 2) 2   (1) k 1
k 1

1 csc 1  ( 1) k 1
.09419755288906060813... 

 2 2

2
2
k 1 k   1
4

 1 / 4 
(4 k  1) 2 (4 k  1) 2  1
1 .09421980761323831942... 



2
2
16 2
k  1 ( 4 k  1)  1 ( 4 k  1)

1479
1
.09432397959183673469... 
 
15680
k 1 k ( k  1)( k  8)
Hk

(k  1)(k  2)
J1058
2 .09439510239319549231... 
.09453489189183561802... 
=
2 .09455148154232659148... 
.09457763372636695976... 
.094650320622476977272... 
2 .09471254726110129425... 
.094715265430648914224... 
.094993498849088801...

.09525289856156675428... 
.095413366053212024...

2
 1
 arccos   
3
 2

1
2
k 1
 log
  k 
2
3
k 1 3 k
k



(1)
(1) k 1
1





k
k
k
2
k 2 2 k
k 1 2 ( 2 k  2)
k 1 3 ( 2k  2k )
3
real root of Wallis's equation, x  2 x  5  0

 ( k )2 k

k
k 1 3  1
 (i )   ( i )
Re (i ) 
2
1
arccsch
4
4

1
4
4k  1
 (2 k  1)!!
 2  


2 
(2 k )!  (2 k  1)(2 k  2)
k 1 
 (2)
1  log k



2
 2 k 2 k 2
 2
2
2
6

5
k 1

1
1037


105
2
1  2k 
  1

k 
k  0 24  k 

2
k 1
k
1
(2 k  7)

(2 k  1)!!
 (2k )!!6
k 1
k
( 1) k

k  1 2 k 
k
 

1 .0955050568967450275...
.095542551491662252...

1
 e Ei ( 2)  eEi ( 1) 

e
 1
 3
10 .095597125427094081792... 
 (1)  
11 .095597125427094081792... 
 (1)   
 4
 3

.09569550305604923590... 

k 1
 ( 3k )
2k  1
Berndt 9(27.15)
  (2k )!!(22k k (12)!!k  1)

 1 3
1 .09541938988358739798...     , 
 2 4
1 .095445115010332226914... 
J393

 4 log 2  2 log 2  2 
8 2 arctanh
J145
1
x2
0 e x ( x  1) dx
k

k
.09574501814355474193... 
k 3
3
1
8
5 


3 2
.09587033987177004744... 
.095874695709279958758... 

2
.09589402415059364248... 
dx
x4
 tanh log x
1
1
 2G  log 2  4    log(1  x 2 ) log xdx
2G 
4 .09587469570927995876... 

0

1
1

 log 2    log 1  2  log x dx

x 
2
0

 1  H ( n) k 
1
4
log       k 
3
 3 k 1 4 
n 1 3

12
k2
  ( 1) k 1 k 
125
4
k 1

4
5 (5)
log x
dx 
.09601182917994165985... 
 4
x x
54
1
.096000000000000000000 =

.096023709153821012933... 
 




 


1 5 
 5 3  5 arc cot 2  2 5  2 5 arc coth 1  5   5  5 arctan h

4 
20 3  5 

F k Fk 1

 
=  k
k 1 4 ( 2 k  1)( 2k  1)

1

.096188073044321714312... 
.096202610816922143434... 
1 .096383556589387327993... 
.09655115998944373447... 
1  (1)  3
log x
(1)  7  
dx
         




64 
8
8  1 1  x4

6
 ( 3) 
2(5)  (2)(3) 

Hk
 (k  1)
k 1

 
=
.096573590279972654709... 
=
=
MHS (4,1)  MHS (3,1,1) 
log 2 1


2
4

( 1) k


k 0 2 k  6

1

k 2
(k  1)
k 1 2

4


1


4
k 2 k j
( 1) k 1
 4 k ( k  1) 
k 1

( 1) k

k  0 ( 2 k  2 )( 2 k  4 )
 k 1
1
 k
k  2 j 1
4
j
J368

=
=
1
k ( k  1)( k  2 )
k 1

 /4
dx
1 x 7  x 5  0 sin2 x tan x dx 
2

=

 (
2
0


1 .09662271123215095765... 
=
x dx
0
1  dx
4 5
x
log x dx
 log 2 x )(1  x 2 )
GR 4.282.4
( 1) k log k

k 1
k 1
2

9
W2

=
5
 log1  x
1
.0966149347325...
 tan
1
1
=
 /4
log x dx
x log x dx
1 ( x  1) 3   0 ( x  1) 3

=
GR 1.513.7
k

2
k 0
k!

k
( k  1)( 2 k  1)!!

2 ( k !) 2
 (2 k  2)! 
J313

1 
1
  6k  5  6k  1
J338
k 1

=
2 .09703831811430797744... 
log x dx
x6  1
0

 2 log 2
3
log 3 2
 2
 1

 2 Li3     2 Li3    
 3
 2
3
1

 
log 2 x
= 
dx
( x  12 )( x  3 2 )
0

e 2
 (k )


2
k
e
k 1 e  1

( 1) k 1 k
.097249127272896739059...   k k
k  1 2 ( 2  1)

e2  3
2k
 

1 .09726402473266255681... 
4
k 1 ( k  2 )!

e2  1
2k
 
2 .09726402473266255681... 
4
k  0 k !( k  2 )
.097208874698216937808... 

1 .097265969028225659521... 

.0972727780048164...
 (k  1)
 2k 
 
k 

1
  4

k 2 k  3
k 1
J277
k 0
 3 log 2 

2k

k 1 k !( k  5)

2

 4
1
4k  1
1
arcsin
2
(4k  1)
4k

 .0974619609922506557...

.0976835606442167302...

.097777777777777777777 =
.097869789464740451576... 
.09802620939130142116... 
.09809142489605085351... 

 
=

 .0981747704246810387...

1 .09817883607834832206... 
.09827183642181316146... 
.09837542259139526500... 
.09845732263030428595... 
1  (4)  7 
( 4 )  3 
        
 8
 8 
37268 
 
=
xyz
   1  xyz dx dy dz
0 0 0

.0984910322058240152...


k 2
1 .098595665055303...
1 .0986122886881096914...


log 4 x
1 x 4  1

5   2 csc  2
(1) k 1
 2
4
k 1 k  4k  2

22
1
 
225
k 1 k ( k  2 )( k  5)

731
1
9e 
 
30
k 1 ( k  1)!( k  6)

i
sin 3k
  Li2 (e 3i )  Li2 (e  3i )    2
2
k 1 k
2


k 1
3
 3 log 2   3

2
12
k 1 4 k  2 k

 ( k )   ( k  1)

( 1) k

2k
k 2



sin 3 k cos3 k
dx
 
  2

32
( x  4) 2
k
k 1
0
2i
csc(log  )  i
   i
 5
 log   
 4

k
166
3
 192 log   k
3
4
k 1 4 ( k  3)

3 (3)
(1) k
1
 1   (3)   3 
4
k 2 k
1 1 1


(k )
k4  k2

 (2 k )   (2 k  1)2
k 1
2k

log 3  2 arctanh
1

2

4
k 0
k
1

( 2 k  1)

 

 
1
= 1  

3
k  1 27 k  3k

2k
 2
= Li1     k 
 3
k 1 3 k
[Ramanjuan] Berndt Ch. 2, 2.2
J 74

=
sin x
 2  cos x dx
0
1 .09863968993119046285... 
 1
3 Li2   
 3

Paris constant
1 .098641964394156...


1
H (2) k

k
2  2
k
k  0 3 ( k  1)
k 1 3
1
.09864675579932628786... 
1 .098685805525187...

.0987335167120566091...

.098765432098765432
=
.098814506322626381787... 
1 si (2)

   log x sin2 x dx
2
4
0
Lengyel constant
2
( 1) k


2
2
24
k  2 ( k  1)


8
log 4 x
k k
  ( 1) k  
dx
81
8
x4
k 1
1


coth
8

5

16

4

1

2

1 .09888976333384898885... 

J401
1
 16k
k 1
2
1
 ( k )( k  3)  1
k 2

5 .09901951359278483003... 
.09902102579427790135... 
.099151048958717950329... 
.099151855335609679902... 
2 .09921712010141008052... 
26

log 2
1  1
1
  k  

2
7
7 k 1 2 k
k 1 28k  14k

1
 (k )  1
 e  1
1  H 


 
k
e
 e  k 2 e
log 2
1 
 
  ( 1) 
4
3 3
4 e



 (k )  1

 k ( k  2)
k 2
k  2

 1  k  1 
k 0
( ( k )  1) 2
2k
k 2


k2
 (k )
  2k
7 .0992851788909071140...   k
2
k 1 2  1
k 1

.09926678198501025408... 
29809 .09933344621166650940... 
 ( 1) k
9
7715
  ( 2 )  5 ( 3) 
1728
1
.099445877615933685923... 

2
2
p prime p p  1
.0993486251243290835...




k
 ( k  5)
k 1
3

2 .099501138291619401819... 
log
3 3

2

.099501341658675...

 ( 1)
k 1
.099586088986234725...
.0996136275967890683...
.0996203229953586595...
 ( 1)
k
k  
k 0
3k  1
 1

  log
 3k
3k 
Prud. 5.5.1.17
 ( 2 k  1)  1
( 2 k )!
k 1
1 .0995841579311920324...


16 2 
( k  14 )!

 4 log 2  
3
9
3
k 1 ( k  4 )! k

Hk
2  4
k
 2 log     ( 1) k
 5  k 1
4 ( k  1)




4

 log 2
8
1 G
 

2 2
1 .09975017029461646676... 
 x tan
3
x dx 
0
3
 e    Ei(1) 
2
 (k  1)! k 
1

csc h  sinh 
2
2
4k 2  1

2
k  0 4k  4

.09963420383459667011... 

 /4
1
k 2

csc 2

3
( 1) k 1 k 2
.0997670771886199093...  2 
   Ei ( 1)  
e
k  0 ( k  1)( k  1)!

1
( k  2 )! 2 k
6 .0999422348144516324...  e  I 0 (1)  
( k !) 2
k 0

 (k )
   1 
.09995405375460116541...  log 
  k 

   
k 2  k
GR 3.839.2
.100000000000000000000 =
1
10
k (k  5)

=
 (k  2)(k  3)
J1061
k 1
8 .100000067076033613307... 
.10000978265649614187... 
.10015243644321545885... 
.10017140859663285712... 
.10031730167435392116... 
1 / 0.1234567891011121314 , inverse of Champernowne’s number

1
 1
 1

Li10     ,10,0   k 10 
 10 
 10

k  1 10 k
k

|  ( k )|( 1)

9k
k 1
 ( 3)
12


2
 (k )  1
3
1
1
 1

 
 
    2   log 1    

2
12
k
k
k 
k 3
k 2  2k


.100391216616618625526... 
.1004753...
93 (5) 5 5
log 4 x

 4
dx
8
128 1 x  1


 a non  trivial
1
2
 , Goldbach’s Theorem
integer power
1 .10055082520425474103... 
3 .10062766802998201755... 
.100637753119871153799... 
.10068187878733366234... 
.10085660243000683632... 

3
10
1 1 1
    x1 xy yzz
3 2
 4 
16
2
2
2
8
dx dy dz
0 0 0

1

 1
Li4     ,4,0 
 10

 10 

2 2
2

 10
k 1
1
k 4
k
1

G 3
    x arctan x log x dx
2 4
0
3  2 log 2

16
1
log x
 ( x  1)
3
dx 
0
( k )
2
1
k 2
2 log 2 13


.10098700926218576184... 
3
36

.10096171...

k
( 1) k 1

k 1 k ( k  1)( k  3)

( 2 k  1)!!
.1010205144336438036...  5  2 6  

k
k 1 ( 2 k )! 3 ( k  1)

 2k 
1
1 .1010205144336438036...  6  2 6   k
 
k  0 12 ( k  1)  k 

.1010284515797971427...

 ( 3)  1
2

1
.10109738718799412444... 
.1011181472985272809...

12 log 2  2 log 2  6 log 2  6 
3

64
2
 
=
7 .10126282473794450598... 
3
0
  coth    3  2 coth  csch  
2
k 4 ( k 4  1)
k  (4 k )  1   4

3
k 1
k  2 ( k  1)


 log (1  x) dx

2
K3 (1)
.10128868447922298962... 
 1
Li3   
 10 
.10132118364233777144... 
 2

1
1

,3,0
k 3  
 10

k
k 1
k
2


 (2)
(1) log k


.101316578163504501886...  
2
k
2
2
k 2
 10
(1) k 1
4
 3

 log  2

k
3
3
k 1 k ( 2k  1)3


1
 ( 3k )
.101560303935709566...
 
  ( 1) k 1
3
11k
k 1 11k  1
k 1

1
2 5
( 1) k
.10159124502452357539... 

csch 5   2
10
5
k 0 k  5

.10148143668599877803... 

.10169508169903635176... 
 log (2 k )
k 2

2 .10175554773379178032... 

k 1
.10177395490857989056... 
G

9

=
k

k!


1
  (12 k  9)
k 1
x dx
0 e 3 x  e 3 x 

2


1
2
(12 k  3) 
log x dx
4
 x 2
x
1

1
( 1) k 1 k
1
1

  k
.10187654235021826540... 
arctan
3
2
2
k  0 2 ( 2 k  1)
2

k H
8 20
4
1 .10198672033465253884... 

log   k k
9 27
3
k 1 4

( k )
.10201133481781036474...  

k
k 1 2

5
(k )
.102040816326530612245... =
  k
49
k 1 7  1
e

e  e 1
( 1) k e k
.102129131964484887998... 


ee2
k  0 k !( k  2 )
.1021331870465286303...

.10218644100153632824... 
.10228427314668489686... 

1613 e
k3
 
k
3
k  0 ( k  2 )! 3

5
 log 2
( 1) k


 
28
14
84
k  0 ( 2 k  1)( 2 k  8)
k


1  log 2
(1)
1

 

2
3
k  0 3k  6
k 1 12k  6k
75 

=
x
7
1
1 .10228427314668489686... 
.10246777930717411874... 
4  log 2
 
3
 /2
k 1
1 2 cos1


2
e
log 2 x sin log x
dx 
1
x2
 sin(3x) log tan x dx
0
e
1

0
x 2 sin x
dx 
ex
1
  2  4 log 2  log     k k 1  (k )  1
2
k 2 2 k

1
.10253743088359253275...  
k
k 1 (10 k )

1
 1
1

.10261779109939113111... 
Li2     k 2   ,2,0
 10 


10
k  1 10 k

37
( 1) k 1
.10277777777777777777 =
  2

360
k 1 k  6 k

1
1  dx

5
 
=  x log(1  x ) dx   log 1   7 

x x
0
1
.10253725064595696207... 
.10280837917801415228... 
2
( 2)

96
16


1
 (4 k )
k 1
2
.102953351968223325475... 


1
k
dx
 x4



 2 3(k

1  
3 (3) 
( 3)  3 
( 3)  1  
 8  24 G  4 2 log 2 
         
6144  
256 
 4 
4
 /4
=

x
0
2
log sin x dx
2
 k)
1
.103171586152707595114... 
1 .10317951782010706548... 
=
1 .10322736392868822057... 
x log x
 1
1  Li2      
dx
 2
0 x2
1   i 
i 
 1 

 1 
 H  1 / 4   H  1 / 4   H   1 / 4   H   1 / 4  
4 2 2 
 2 
 2 
2 


 ( 4 k  1)
k
  4

k
2
k 1
k 1 2 k  1

2k
1
G
2 2 log 2  35 (3)  
16
k 1  2k  3
 k
k 



2 .10329797038480241946... 
.10330865572984108688... 

5
  5k  (3k )  1

3
k 2 k  5
k 1

( 1) k
106
1
 8 2 arctan
  k
15
2
k 0 2 ( 2 k  7)

1 .10358810559827853091... 
 1  log (2 k ) 
k 1
.10359958052928999945... 
 (3)
4
1
=

0
2 .10359958052928999945... 
=


 2  2 (3)  2 
1
2
log 2 x
dx 
x4  x2

x 2 dx
1 e3x  e x
2
x log x
dx
1  x2
GR 4.262.13

7 ( 3)
1
 2  ( 3)  2 
3
4
k  0 ( 2 k  1)
1
log 2 x
0 1  x 2 dx
GR 4.261.13

=
.10363832351432696479... 
1 .10363832351432696479... 
.10365376431822690417... 
.10367765131532296838... 
.10370336342207529206... 
x 2 dx
1 e x  e  x

k2
3
k 1
 1   ( 1)

( k  1)!
e
k 1

( 2 k  1) 2
3
  ( 1) k
( k  1)!
e
k 1


cos ex
dx
e  
2e
1  x2
0
10 1
 
9 4
2
k5
( 1)

( k  1)!
k 0
k
AS 4.3.146
2
1
 (2 k  1)!!



k  1  ( 2 k )!!  2 k  4
  log 2 2

2 


   log 2 
2
 log 2
2

2
8



4

=
x e
2  x2
log 2 x dx
0
.10387370626474633199... 


=

11  2 cot  2  coth  2


24
16

k
k 2
4
1
4
 4  (4k )  1
k 1
k 1

5  8 log 2  2 log 2 2
H
  ( 1) k 1 k
4
k 3
k 1
k

G 1
( 1) k
.10399139854430475376... 
  
2
2
4 8
k 1 ( 4 k  1)

dx
.10403647111890759328...   6
x  x3  1
1
.1039321458392100935...

1 .1040997522396467546...


 ( 4k )
 k (k  1)
k 1

3
dx
.10413006886308670891... 
  2

( x  2) 3
64 2
0

1
1 .10421860013157296289...   k
k  0 3 ( 3k  1)
.10438069727200191854... 
2
24
 log 2  1 
k 1


 
=
2
k 1

 4k
k
1

2
( 2 k  1)

 ( 1)
k 3
Hk
k (k  1)(k  2)
log 10!

 2
k2
.10452866617695729446... 
  2
sinh  2
k 1 k  2
 2
  log 2 3
x log x sin x
.10459370928947691247... 
 
  
dx
2 8
4
4
ex
0
15 .10441257307551529523... 
.10459978807807261686... 

 
=

1  k!(k  1)! 
1
 


3 3 2 k 1 (2k  2)! k 1  2k 
 (4k  2)
k 

dx
1 x5  x 4  x3 
(1) k 1 k

k 1 ( 2k  1)!( 2k  1)

.1046120855592865083...

si (1)  sin 1  2
k 1
( k )
2k

1

 
=
  x log x sin x dx
0
  1/ 8  1 
1  
1 1 
 e  I 0    I1     1 
 1 F1  ,2,  1 
2   2 4 
2 
 8   
 8

(k  12 )!(k  12 )!
 
= 
k 1 ( 2k  1)!( k  1)!

8
9
H
.10469603169456307070... 
log   ( 1) k 1 kk 
9
8
8
k 1


 1  dx
1 .1047379393598043171... 
erf 1  erfi1   cosh 2  2 

x  x
4
1
.10464479648887394501... 




 
=
.104742177484448455438... 
 1  dx
2 cosh 4  3
x x
1
1  2
 13  
5
 4 20  14 2   (1)     (1)    
256 
 16  
 16 



 
=
log x dx
4
 x 4
x
1
3814279 .1047602205922092196...


e
ee 

1 .104780232756763784223... 
k
k 2
2
Hk
log k

.10492617049176940466... 
 ( 1) k 1k 3  (3k )  1) 
k 1
.10493197278911564626... 
617

5880

k 3 ( k 6  4 k 3  1)

( k 3  1) 4
k 2

1
 k ( k  1)( k  7)
k 1

k2
.10498535989448968551...  71  43 e  
k
k 1 ( k  1)! 2 ( k  3)


 (k )
1


.105005729382340648409...  

 k Li3  1

3


k
k  2 ( k  1)
k 1



 (2 k )
1 

.10500911500948221002...  log
  k    log 1 

 16k 2 
2 2
k  1 16 k
k 1
1

 
1  x x2
dx
=  

2
1  x 1  x log x
0

1 .10503487957029846576... 

 1  
k 1
2
1


k ( k  1) 
GR 4.267.5
.10506593315177356353... 
7
 ( 2) 
4

1


4
2
k 2 k  k


 
=
  (2 k )  1


k 2

=
ω a non  trivial
integer power
5 .1050880620834140125...

1

2
 1

 ( k )  1
k 1
.1052150134619642098...
.105263157894736842

=
.10536051565782630123... 

 
1 .1053671693677152639...
=

1 .10550574317015055093... 

 .1055728090000841214...

.1056316660907490692...

.10569608377461678485... 
.105762192887320219473... 
2

k  ( k  2)  1
(k )
2
k 2 k


2
5 3 log 2


8
4
.10516633568168574612...   10
.10518387310098706333... 
k 2
 k ! 2
.10513961458004101794... 
1 .10517091807564762481... 
k 1
k 1
1
 k ( k  1)
13
8

.10513093186547389584... 
 ( 1)


2
k 2
k
1
( k 2  1)
e1/10
sin 1

8
( 1) k 1 k 3


k 1 ( 2 k  1)!

 /2

  ci      sin x log x dx
 2
2
19
0

 1

  ,1,0 

10 
k
k 1

1
1

2 arctanh
  2 k 1
19
( 2 k  1)
k  0 19


1
 (k )
  2 k

k
2
2
k 1 ( 2  1)k
k 1

k ( k  2) Hk
( 3)   ( 2) ( 3)  2 (5)  

( k  1) 4
k 1

2
( 1) k 1 ( 2 k  1)!!
1
 
( 2 k )! 4 k
5
k 1


 ( 3k )  1
k




k
3
2
k 1
k 2 2 k  1

2
1
  x 3 e  x log x dx
4
0
log 10  log 9 
1
 10
k

i ( 3)
 1
1 
 (1  i )   ( 3) (1  i )  i  


4 
( k  i)4 
6
k 1  ( k  i )

.10590811817894415833... 
=
.10592205557311457100... 
1 .10602621952710291551... 
.10610329539459689051... 
.10612203480430171906... 
2 log 3
2 log 2
 2 log 2 log 3  log 2 2 
 log 2 3
3
3

( 1) k 1 kH k

k
k 1 2 ( k  1)

8
1
4 log 2    k
3
k 1 2 ( 2 k  6)


4
1 

sinh    1 

4
16k 2 

k 1 
1

3
 log 2 26  
1  dx


  log 1  2  6 

10
5
75
x x
1


.10618300668850053432... 
 (1)
k 1
 (k  1)  1
k2
k 2


1
.10620077643952524149...   4
    ( s)  1 ds
k  2 k log k
4
.10629208289690908211... 


4e
2

cos x
dx 
2
4
x
0
 /2

 
=

 cos(2 tan x) sin
2
x dx 
GR 3.716.4
0

2 .10632856926531259962... 
 k  2
( k ) 1
k 2
 1 
log 2 log 2 2


2
2

( 1) k kH k
= 
k 1
k 1
.10634708332087194237... 

 
2
1
log(1  x )
 1
.10640018741773440887...  log 2 
 Li2     
 2
x3
12
0
1
.10642015279284592346... 
 i  (1 ) ( 2  i )  i (1 ) ( 2  i )   ( 2 ) ( 2  i )   ( 2 ) ( 2  i )  
8


k ( k 2  1)
k 1 2
=  ( 1) k  ( 2 k  1)  1   2
3
k 1
k  2 ( k  1)

24
1  11k  11k 2  k 3
.10649228722599689650...  24 (5)  14 ( 3) 
 ( 2)  1  

5
( k  1) 5
k 2
2


 
=
 ( 1) k  ( k  1)  1
k
k 1
4
.10666666666666666666
=
.10670678914431444686... 
x
4
arccos x dx
0
1  3
 3

csch

6
12
2

.106843054835371727673... 
1
8

75

log 2  3
3
( 1) k 1


2
k 1 4 k  3

  1 log26  15 3 log2  3  
6
(1) k 1 2 k k

k 1  2k 
 (k  1)(2k  1)
k 


 
=

.106855714568623758486... 
 ( 1)
k
 (k )  1
2k  1
k 2



 
k arctan
k 1
1
1
1 
3k 
k

( 1) k
1
 10
5

csch 
  2
10
20
2
k 0 2 k  5
2 k 1

2
k 2
  (1)
1 .10714871779409050301...  arctan 2  arcsin

2k  1
5
k 0
.10691675656662222620... 


1 .1071585896687866549...

 H  ( x)  1)
x
2

.10730091830127584519... 
1 


2 8
( 1) k 1


2
2
k 1 ( 4 k  1)


=
=
=
1 .10736702429642214912... 
.10742579474316097693... 

1


2
k 1 16 k  1

( 1) k

k 0 4 k  6

1
1 x 7  x 3

k 1
GR 0.232.1
16 k
28
1  4 log

.1076723757992311189...
 10  6
4

5

1


k
k 1 5 k ( k  1)
one-ninth constant
e 

k
 ( k  2)! 2
k 1


( 2 k )
3
.1076539192264845...
3 .1077987001308638468...
GR 0.237.2
 1
e I 0   
 2
k
( k  12 )!

2
k  0 ( k !)
(1) k 1

k
k 1 4 ( k  1)

.10801497980775036754... 

4
coth

2
 
i
i 
1


1   1     1    

 2
2
2

k 1
( 1) k 1
 
 (2 k )   (2 k  1)

3
4k
k 1 4 k  k
k 1

1  e 2 e sinh 1
(1) k 2k

 
8
4
k  0 ( k  3)!

( 1) k
3
3 1
log 
  k 2
2
2 2
k 2 2 ( k  k )
k

3
( 1) k  2 k 
 
 
k
k
16
k  0 12


 
=
.10808308959542341351... 
.10819766216224657297... 
.1082531754730548308...


32
(1) k 1
 
2
9
k 1 ( k  3 / 2 )

1
1 .10835994182903472933...   k

k  1 4  ( k  1)  1

H
1 .10854136010382960099...   k k
k 1 2  1

2

1
.10874938913908651895... 
 1  4    e  x sin 2 x cos2 x dx
16 
e 
0
.10830682115332050466... 
4G 
( ( k )  1) 2
k2
k 2
27  4 3
.10905015894144553735... 

48

.10900953585926706785... 
.109096051791331928583... 
.10922574351594719161... 

71  6
108
2


1  dx
3 5 
x
 log1  x
1

1

2 
k  1 k ( k  3)
log  1 7 (3)
 

2
4
4 2
k 1
x
2
log(1  x ) log x dx
0
 ( 2k )

4
1
k
k ( 2 k  2)
1
.10928875430471840710... 
x 6 dx
0 1  x12

7
k
  ( 1) k 1 k
64
7
k 1

1
k
 1
1 .10938951007484345298...   sin
 ecos(1/ 2 ) sin sin 
 2
2
k 1 k !
.109375000000000000000 =

( 1) k 1
1
 2

csch  2   2
12
4
k 1 k  2 k  3
k

1

 2k 
4 2 5
( 1)
  k
.10947570824873003253... 
 
6
k  1 4 ( k  2)  k 
.10946549987757265695... 
( ( k )  1) 2

2k
k 2

.10955986393152561224... 
2
.109575787818588751201... 
432


12 3

1  (1)  1 
 2
     (1)    
 3 
 6
144 
( 1) k

2
2 
k  0 ( 3k  1) ( 3k  3)


 
=

log 2 x
1 2 x3  x 2 dx
.10960862136441250153... 
 1
2  4 Li3    
 2
.10965469252512710826... 
1
Ei (  e2 )  Ei (1)  

2
( k )
.10967200...
  2

k 2 k  1

1
2 .1097428012368919745...  
2
k  2 k log k
9
1
.109756097560975609756 =


82
2 cosh log 9
1
e
e2 x
dx 
0


.10981384722661197608... 

.1098873045414083466...

.109944999939506899848... 
 

log 2 

4




 ( 1)
k
e 2( 2 k 1) log 3 

7
(1) k 
1

 2

12 k  0 k  5 k 1 4k  14k  12
arctan 2

2
 
 /4
J943
k 0

x
1
6
dx
 x5
2
sin x
 1  sin
2
0
 
x
dx 

4

 5 log 512 11  5  11log 11  5  2 7  2 5 log 11  5  3 5 log  
3
51 5 

1

(1) k 1 Fk Fk 1
= 
4 k k (k  1)
k 1



 


.11005944310440489081... 
1
k
k 2
.11008536730149237917... 
3
3
log
2 2
k
k 1
64

3 ( 3)  2
( 1) k

 2 log 2  3   4
3
4
12
k 2 k  k
1  ( 1)  1 
 3
 5
 7   log(1  2 )
.11037723272581821425... 

      ( 1)     ( 1)     ( 1)    
 8
 8
 8
 8
256 
4 2
.11030407191369955112... 
(1) k

2
2 
k  0 ( 4 k  1) ( 4 k  3)


 
=

9 .1104335791442988819...
83

1 .11062653532614811718... 
1 .11072073453959156175... 

 ( 3)
( k )
  4 
( 4)
k 1 k


(2 k  1)!!
 1 

k
2 2
k  1 ( 2 k )!!2 ( 2 k  1)

 
=
 ( 1)
k 1
  4 k  3
( 1) 

4k  1 
Titchmarsh 1.2.13

 2k 
1
  k  8 (2k  1) 
k 1
k
k 1

k 1
=
(1) k / 2 

k  0 2k  1
=

( 1) k 

1



2 k  3
k 0 
J76, K ex. 108c

Prud. 5.1.4.3


=
x
0

 
=

0
 (1  x
0
 /2
=
Seaborn p. 136
dx

2
x 2
1
=
dx
1
4
dx
0 2 x 2  1 


0
x 2 dx
x4  1
dx
2
) 1  x2
d
 1  sin
0

2



1 1
 (k )

 2 
log       k    log 1   
 3k  k
 3 3
k 2 3 k
k 1
k

( 1)
.11083778744223266...
  3
k 0 k  4


1
 ( 3k )
.11094333469148877202...  
( 1) k 1


3
10 k
k 1 10 k  1
k 1
.11074505351367928181... 
.111001751290904447718... 
9 ( 4 )  6 ( 3)   ( 2 ) 
.11100752891282199021... 
4

k2

4
k 1 ( k  3)
65

16
log 3 2
3

( k )
.11100942712945975694...  
3
k 2 k ! k
.11100821732964315991... 
.11105930793590622477... 
1 .11110935160523173201... 
.11111111111111111111
=
2
3
 1
e Li2   
 e
=
.11118875955726352102... 
.1112477908543131171...

.111387978350712564526... 
.1114472084426055567...


 
=

 
=

 
=
1 .1114472084426055567...

 

=
.11145269365447544048... 
k 0
k
log x
1 x 4  
(k )
3
k 1
e

1

9

 

1
k


 /2
 log(sin x) sin
k
2
1


(k )
9
k 1
k
1
1

16 G  2 2 log 2  21 (3)  
64
1
12 (3) 
 4 log 2  5 
 
8
2 
 

2
4
 
 log
x cos x dx
0
(k )
6
k 1
1
( k  1) 2
9 44


4
9

 /4
x
2
tan x dx
0
 ( 2k )  1
 k (k  1)(k  2) 
k 1
(1) k 1 k 2
 (k  1)  1 

k
k 1 2 ( k  1)

1 
1 
1 

  1 
   1 
  1

2 
2
2 
1
1 



 1    cot
  1 


2
2
2
2
1  1 
 1 
  H 

  H
2  2 
3  



1
 ( 2 k  1)  1




3
2k
k 2 2 k  k
k 1
1 
1 
1 

    1 
   1 
 
2 
2
2  



1
 ( 2 k  1)
 

3
2k
k 1 2 k  k
k 1
 2 log 3 2  2 log 2
 1 3


 2 Li3      (3) 
 2 2
6
3
3


 
=
.11157177565710487788... 

 
=
.11157214582108777725... 
.11163962923836248965... 
1 .1116439382240662949...


 
=

 
=

 
=
log 2 x
1 (x  2)(x  1)2

1
5
1
1

log  arctanh   2 k 1
2
4
9
( 2 k  1)
k 0 9
i 


Imi log 1    
 2 


(1) k 1 k
i  (1) 
i
 i 
 1     (1) 1     
 (2k  1) 
12 
9k
 3
 3 
k 1

1


4
k 1 k  5





1
 3i   3  3i  3    1  i 3      1  i 3   
 2
 2
2 
2  
3 3  3i 


1
1    1  (1) 2 / 3  (1)1 / 3   1  (1)1 / 3   (1) 2 / 3
3


1
1


( 1) k 1  (3k  1)  1)  


2
1
2 k 1
k 1 k  k







 


1   ( 1) k 1  (3k )   (3k  1))
k 1

( 1) k
.11168642734821136196...  

k  0 5k  6

.1119002751525975723...

 1
Li4   
 9

.11191813909204296853... 
x
7
1

dx
 x2
1
9 4
k 1
  ( k )  1
k
k
5
k 2


2
 ( 2 k  1)  1




3
2 2
)
2k
k  2 k (1  k
k 1


(   2)
k (2 k )
16k 2
 
 
.11207559668468037943... 
2
2
32
16 k
k 1
k 1 (16 k  1)

1 
i
i 
1

.11210169134154575458...      1      1     
2 2
  
2  
k  1 k (1  k  )

( k )
.11211354880511828217...   k 1

1
k 1 2
.11196542976354232023... 

1 .11225990481734629821... 
 H  (k  1)  1
k 1

6 .1123246470082041...

( 2)
k
e1/ k

k 1 Fk

4 .11233516712056609118... 
=
5 2  3 (2)  d (k 2 )



 (4) k 1 k 2
12
Titchmarsh 1.2.9
1  p 2
p (1  p  2 )2

.1123768504562466687...

.11239173638994824403... 

 
=

 
=
2 .11259134098889446139... 
.11269283467121196426... 

 
=



1 1 1
=

xyz
dx dy dz
2 2 2
y z
  1 x
0 0 0
.11270765259874453157... 

e
8
1
 1  dx
 1  dx
   sinh 2  9   sinh  5
x  x
 x x
2e 2
21
1
1 2
  21  3 2 csc  2 
24
1

  2
sec
 csc
  sin  2  3 2  21sin  2 
48 
2
2

( 1) k

4
2
k 2 k  2 k

21/ k

4
k 1 k

3 ( 3)
( 1) k

 
3
32
k  0 ( 2 k  2)

1
1
k
  Li3 (i )  Li3 ( i )    cos

2
2
k 1 k
 1
Li3   
 9

2


9
k 1
1
k3
k
log 2 G


16 8
2
2

( k  2)  1
.11281706418194086693...  
( k )
k 2
.112741986560723906051... 
.11281950399381730995... 
. 11292216109598113029... 


1
1
log( 2 e  1)  1  
2
e
1 2


4 72


1
 (2k  1(4k  1)(4k  3)
k 0
1

0
1  e x  2
2  e x
H k H k 1
 k (k  1)(k  2)(k  3)
k 1

1
  k
  , ,1   k
4 .11292518301340488139... 
 
 k 1 

3
4
( 1) k 1 k
.11304621735534278232... 
 3 log   k

4
3
k 1 3 ( k  1)
1

1 .1131069910407629915...

  ( x)  1 dx
3/ 2
dx 
2

1 .11318390185276958041... 
.11319164174034262221... 
1
 k  k  k  k4  k3  k2  k 1
k 0
 4
 log   
 3
k

.11324478557577296923... 
8
7
6
5

   ( jk  4)  1
j 1 k 1
4 .11325037878292751717... 
e
1 .113289355783182401858... 
(1) k


k  4 S1( k ,4)
2


3 log 3  3 1
 (k  1)  1


 
2
2
5
6k
k 1

5 
5
Hk k

 1  log    ( 1) k 1 k
36 
6
5
k 1
k

( 1)
  k 1
k 2 k

log k
 
k
k 2 k !2

1
3


J104, K ex. 110e
 2 log 2  
2
2
2
k 1 k ( 4 k  1)


1
1
= 
  k
k 1 k ( 2 k  1)( 2 k  2 )
k 1 2 ( k  1)( k  2 )
.113513747770728380036... 
.11356645044528407969...
.11357028943968722004...
.11360253356370630128...
.11370563888010938117...
2 log 2 
k 1


3
k 1 4 k  k

(1) k


2
k 2 k  k

=
=

=
dx
1 ( x 2  x)2 

 ( 2 k )   ( 2 k  1)
k 1
4k


e
x
1
dx

(e x  1) 2
|  ( k )|( 1) k

9k  1
k 1

16
( 1) k
6
 

e
k  0 k !( k  4)

.1138888888888986028...

.11392894125692285447... 
e
log 3 x
1 x 2 dx
(1) k

k  2 k ( 2k  1) log k

.1139565939558891...

.11422322878784232903... 
9
7  23/ 4 


csc 23/ 4   csch 23/ 4 
112
224


( 1) k

4
k 2 k  8

=
.11427266790258497564... 
=
.1143602069785100306...

 

=
2 .11450175075145702914... 

 
=
.114583333333333333333 =
.11466451051968119045... 
.11467552038971033281... 
.11468222811783944732... 
.11477710244367366979... 
3 .11481544930980446101... 
3 .11489208086538667920... 
.11490348493190048047... 
.11508905442240150010... 
.11512212943741039107... 

 


 


 
=
4 8 log 2  8 log 2 log 3  2 log 2 3  3 log 3  1

Hk

k
k 1 4 ( k  2 )

1
 1
Li2     k 2 
 9
k 1 9 k
2
1 
6 Li2   
 log 2 3
 3 3

1
Ei (1)  Ei ( 1)  2 

k  0 ( 2 k  1)!( 2 k  1)
 1   3 3  1 
2 SinhIntegral (1)   2i si (i )  2 HypPFQ  ,  , , 
 2   2 2  4 

11
1
 
96
k 1 k ( k  2 )( k  4 )

k
22
4 log 2 2  4 log 2 
2  k
2
3
k 1 2 ( k  2 )



1
(1) k 1
csch

  2
2 2 6
6
k 1 6k  1

( 1) k 1 (2 k  1)!!
2 log 2  5  4 log 2  
(2 k )! 4 k k
k 1


1
1 1
1
cos  sin 
16
2 8
2
( 1) k 1 k 2

k
k 1 ( 2 k )! 4

2 tan 1

1
sinh k
 1 
 Ei (e)  Ei     1  
 e
2
k 1 k ! k
J 2 (1)

(1) k

3
k 2 k  1

1
3
4
6
  cot
 4 cos log sin  
  cot
10 
5
5
5
5
1
8

12
2 
  4 cos log sin  4 cos
log sin  
10 
5
5
5
5
1
16
2 
log sin  
  4 cos
10 
5
5
 2
9
3
 4
           
5
 10 
 10 
 5


 
=
1
 (5k  1)(5k  2) 
k 1

log(e  1)  1
log x
dx
 
e
x e2
0 (  )
1
.11524253454462680448... 
.11544313298030657212... 
.1154773270741587992...

5
log 3 2 1 2
 1
 1  7 (3)
 

 log 2 log 3  (log 2) Li2     Li3    
 2
 2
3
2
8
log 2 (1  x )
= 

x ( x  2)
0
1

 
.11552453009332421824... 
log 2

6


 
=
2
log 2 x
1 x 2  1 dx

 24k
k 1
1

2
 12k

1
62
k 1
k
k

dx
 ( x  1)( x  2)( x  4)
0
3 2
.11565942657526592131... 
256
3
2 3
.11577782305257759047...    log 2  6 
log

2
2 3

1
 
=
 log(1  x ) dx
6

 
1  dx
6 2
x
1
1  dx
6  2
x
1
1 5 
1
1
 
log 2 
arccsch 2 2 

2
2
2


k


(1) (2k )!!
1
(1) k 1
=  2 k  2
 

(2k  1)!! 2k  2
k 0 2
k 1  2k  2
4   k
k 

J143
( 1) k

2 k 1
( 2 k  1)
k 1


 (k )
 (2k  1)  1
.11593151565841244881...  log 2     k  

k 1
k 1 2
k 1

 (2k  1)

=  k
k 1 4 ( 2k  1)
.11591190225020152905... 


 log1  x
 log1  x

0
.1157824102885971962...


1
1
arctan 
4
2

2
1  1


   k log 1  2    

k  k
k 2 

=


 

 dx
x
  cos x  1  2 x 

j3
0
1 .1159761639...
1
=
J311
.11621872996764452288... 

 
1 .1165422623587819774...
=

.116565609128780240753... 
.1165733601435639903...

.11683996854645729525... 
.11685027506808491368... 
.117128467834480403863... 

 
=
.11718390123864249466... 
2

9  27


4
12

sin 3k

k3
k 1

2
i
Li3 (e  3i )  Li3 (e3i ) 

2
1  1
  sin x 4 
   cos  
dx
3  4
8
x4
0
7 ( 3) log 2 1

 
4 2
2
4
 (2 k )

4
GR 1.443.5
( 2 k  2)

 2k 
k
arcsinh 2
1

  ( 1) k 1   k
 k  2 (2 k  1)
2 2
2 3
k 1

( 1) k 1 Hk
7
8
log
 
8
7
7k
k 1


2 1
1
1

J247, J373
 


2
2
2
16 2 k 1 (( 2 k  1)( 2 k  1))
k 1 ( 4 k  1)
1
3
6 (3) (3) (3)  6  (3)    2 (3) ( 3) (3) 
3
 (4)
k 1
k



 (k ) log3 k
k 2
k3

5 2
 32 
3
( 3)  24 log 2 

k
k 1
3
1
( 2 k  1) 2
15
( 1) k
 
128
3k
k 1

 13
( 1) k
.11735494335470499209... 

 
12 90
k  0 ( 2 k  1)( 2 k  7 )

2
1
1
.11736052233261182097... 
 arctanh   k
3
2
k  0 4 ( 2 k  1)

1
(1) k 1
8   3 csc  3   2
.11738089122341846026... 
6
k 1 k  4k  1

k
4
.117187500000000000000 =

.11738251378938078257... 
8 .11742425283353643637... 
2 .11744314664488689721... 

2G  1 1
 
2
3
4
12
 4 ( 2) L ( 2) 
2(4) 

2
k 1
.117571778435605270...
.1176470588235294

=
(3)  (2) 
2
17
2
 (2 k  1)!!
1




k  1  ( 2 k )!!  2 k  3

1/ k 4
( 4 )
4

r(k )

2
k 1 k


3 

Hk
 (k  2)
k 1
2
J385

1
( 1) k 1
1 
  k
  ,1,0 

k
9 
k 1 9 k
k 1 8 k

1
1
 
= 2 arctanh

 2  2 k 1
( 2 k  1)
17
k  0 17
1  (k  1)(k  2)
 (k  2)  1
1 .11779030404890075888... 

2 k 1
k

9
.11778303565638345454...  log 
8

.11785113019775792073... 
.117950148843131726911... 
1 .1180339887498948482...

 

=
.1182267093239637759...

.11827223029970254620... 

 
=
.11827410889083245278... 

 
=
.118333333333333333333 =
2 .11836707979750799405... 
.118379103687155739697... 
.118438778425057529626... 

 

=
1
6 2

 /4
 /4
0
0
2
 sin x cos x dx

sin 3 x
dx 
tan x
1  1
i
 i
Li2      Li2    Li2    
2  4
 2
 2

 2k  1
5
1
      k 
2
2
k  0  k  20

1

k 1 1  F2 k 1

( 1) k
3
 cos1  sin 1  
2
k 1 ( 2 k )!( 2 k  2 )
1
1  (1   ) sin 1  (cos1  1) log(2  2 cos1) 
2

cos k

k  1 k ( k  1)

G log 2
( 4 k  1) ( 2 k )

 

k
4

k 1 16 ( 2 k  1)

1
1

2 k  arctanh
 2 arctanh 


2k
4k 
k 1

71
1
 
600
k 1 k ( k  1)( k  6)

1
k 2

log

k
k 1 k
7 4
  Li4 (i )  Li4 (i )
5760
5 3
13 (3)
1  ( 2)  5 
 1 
    ( 2 )    


36
432 
162 3
6
 3 
k

(1)


3
k  0 (3k  2)

.118598363315566157860... 








72 5
11  5
64
 14  4 5 log
 11  5 log
 log
 6 3  5  
2

8
27
(1  5 )
53 5 


1
K148

(1) k 1 Fk Fk

k
k 1 4 k ( k  1)


 
=
.11862116356640599538... 
1  e,1 

x
e  x
e dx 
1
( 1) k 1 (2 k  1)!!

.11862641298045697477...  1  log 1  2  1  arcsinh 1  
k  1 ( 2 k )! ( 2 k  1)



J389
( 1) k 1  2 k 
 

k
k  1 4 ( 2 k  1)  k 

=
log1  k 2 
.11865060567398120259...   Hk 1 ( ( 2 k )  1)  
1 k2
k 2
k 2

k3
1 .11866150546368657586...  2 e    3  Ei (1)  
3
k  0 k !( k  1)


2 2  1 
log x
dx 
   
2
8
 4
(
1

)
x
x
0
1
4 .11871837492687201437... 
.11873149673078164295... 

4

2

3
( 1) k


k  0 2k  5

1
  1
.11877185156792292102...     log 2 

 8 2
2

5 .11881481068515228908... 

0

1
dx

6
x  x4
 /4
 tan
4
x dx
0
x 2 arctan x
dx
x 1
2
 ( k  1)!! 
k 0


k
1

GR 4.241.11
( k  1)!!

k!
k 0
 4 51

  ( 3)  3 
15
8

7 .11881481068515228908... 
.1189394022668291491...




1
log 3 x
dx 
x4  x3

x3
0 e2 x e x  1 dx

.11899805009931065100... 
  (2 x  1)  1 dx
1
.11910825533882119347... 
.11911975130622439689... 
 5
36

2
144
2 sin 1 1
 
2
e
1

log 2
   x 2 arccot x log x dx
18
0
e
log 2 x coslog x
dx
1
x2

1
( 1) k 1



e2  1
e2 k
k 1
.11925098885450811785...   ( 4 )   (5)  2
.11920292202211755594... 

cos(ex )
dx
ex
0

.119366207318921501827... 
3

8

.1193689967602449311...

9 .11940551591183183145... 
.1194258969062993801...

.11956681346419146407... 
.11973366944845609388... 
.11985638465105075007... 
.11995921833353320816... 
.12000000000000000000
=
1 .12000000000000000000
=
.12025202884329818022... 
.12025650515289120796... 
.12041013026451893284... 
.12044213230101764656... 
.1204749741031959545...

1 .12049744137026227297... 

 
=
.1205008204107738885...

.1205922055573114571...

 ( 1)
k 1
1 ( k )
2k  1
k 1
4
   4
 2  8G
144


1
 (12 k  9)
k 1
2
( 1) k 1 k

k 1 ( 2 k  2 )!

k
( 3)   ( 4)  
4
k 1 ( k  1)

( 1) k 1 k
1
1
sin  
k
4
2
k 1 ( 2 k )! 4

Ei (1)
( 1) k 1  ( k  1)
  
e
k!
k 1

3
 (k )
  k

25
k 1 5  1

FF
28
  k kk  2 
25
4
k 1

( 1) k
1
1 1
4
arctan  log   2 k 1
2
2 2
5
k ( 2 k  1)
k 1 2

 2 k  ( 1) k 1
73 3
   k
15
k  1  k  2 ( k  3)
cos1 
sin 1

2

2

1
Hk
( 3) 

 
2
12 2
k 1 k ( k  1) ( k  2 )

( 1) k
 HypPFQ {},{1,1,}, 1

3
k  0 ( k !)

( 1) k k
e
e log( e  1)  e 
  k
e  1 k  0 e ( k  1)

H k 1




coth  (1  i )  

2
2
2
k 2 k  1
i
2
2
 (1  i )   (1  i )   (1) 1  i    (1) 1  i 
 

4
1
 (1  log 2)
   x arcsin x log x dx
8
0

2 log 2
47


5
300

1


k  0 ( k  1)( k  5)
( 1) k 1


2
k 1 k  5k




 
1  dx

=  log 1   6 

x x
1
1 .12065218367427254118... 
1 .1207109299781110955...

 

3

1
x
4
log( x  1) dx
0
( 1) k

2
k 0 k  3

 log 2 

1
(4 k  1) (2k )



2
8k
4 2
2 2
k 1 8k  8k  1
k 1


1 
6k 2
 1
=  





2
2
 2
8k 2  1
k 1 2 k  1
k  1 ( 2 k  1)(8k  1)

.12078223763524522235... 
tan



 
1
4
 3
  log   log    
 log
 2
2

 2 


 
=
1  dx
1
   x


 2 e  1 e 2 x x
0


 
=
 x
 sinh  2  xe
2
x
0
GR 3.427.3
dx

cosh x
GR 3.553.2

3 ( 3)
( 1) k 1
 12 log 2  10   3
3
2
k 1 k ( k  1)

1
.12092539655805535367...   4
k 2 k  6

3
( 1) k
.12111826828242117256... 
 
3
256
k  0 ( 4 k  2)


1  2  k 
( 1) k
.12112420800258050246...  
 
k
k2
k  1 1  2 
k   2
.1208515214587351411...

1
1
coth  2



6 24
22
 5
.12114363133110502303...  log   
 6
.12113906570177660197... 

 .1213203435596425732...


 
=

k
k 1
 /4
0

.121606360073457727098... 

2
x dx
 (2 k  1)
4k  1
4 (2) g 2
1
1

  (1)   
9
2
9
 3
k 1
1 .12173301393634378687... 
1

 
=

0
1
(k  2 )
( 1) k 1 k  2 k 
 

k
k  0 4 ( k  1)  k 

3
2 
2
 sin x tan
2
log x
dx 
x3  1


1
x log x
dx
x3  1
2
.12179382823357308312... 
 ( 3)
2
.12186043243265752791... 
4 log
3
3
1  1


    ,1, 3 
2
2
2  2

(1) k 1


k
k 1 2 ( k  2)


(1) k


k
k  0 2 ( 2k  6)


1

3k 2
k 1 k  k
3  
 k 
2
 3


14
a(k )
1 .12191977628228799971... 
( 3)   4 
Titchmarsh 1.2.13
15
k 1 k
1 
i
1  i 
 i 
 i 
.12200602389514317644...     1     1      H    H     
2   3
2   3
 3 
 3 

 
=
 
(1) k 1 (2k  1)
= 

9k
k 1
3


.12202746647028717052... 

 3
6
9 log 3
 6 log 2 

2
1 
.12206661758682452713... 
 csch 2  
8 4

sin x cos x
 
= 
dx
ex  1
0


1
 log1  ( x  2)
0
3

 dx 

( 1) k 1


2
k 1 k  4


4 3 3
1
1
   ( 2 )    2
1 3
9
 3
k  0 (k  3 )


( 1) k 1  ( 3k )
1
.122239373141852827246...  
  3
k
9
k 1
k 1 9 k  1

 (k )
1 .12229100107160344571...   ( 4 ) (5)   1 5
k
k 1

 (k )
=  1 4
k
k 1

( 1) k

.12235085376504869019...  1 
 log 2   2
2
k 2 2 k  k

( k )
2 .12240935830266022225...   k  2
k 2 k

( 1) k 1
2 1
.12241743810962728388...  2 sin
 
k
4
k 1 ( 2 k )! 4

6
( 1) k 1 k
.12244897959183673469... 
 
49
6k
k 1

 ( 1) k 1 
1/ 6
1 .122462048309372981433...  2
  1 

6k  1 
k 1 
55 .12212239940160755246... 
26 (3) 
HW Thm. 290

1  17
1

2 arcsinh 2    2 log 2
4
4
.12248933156343207709... 
1
1
1 
( 1) k
arctan   k

2
4
8 k  0 16 (2 k  1)
x
  8 log 2  11
1
 log x dx 
x
2
.12263282904358114150... 
36


(1) k 1

k 1  2k  k 2
  4 k
k 
.122479299181446442339... 
 
=

1
.1228573089942169826...

.12287969597930143319... 
.1229028434255558374...
.1229495015331819518...
.1230769230769230769
1


2

   x 2 log 1 

0
3
dx
 x 1
log( x  1) log x
dx 
x4
( 1) k 1 k 2


k
k  0 ( k  1)! 2
log 2 2 log 3 2  ( 3)
log 2  1 



2
3
4
7
2 
e

log 2 (1  x )
0 x (x  1)2 dx 
1

sin 2 2
( 1) k 2 k
 
1
2
k  0 k !( k  2 )

5 5
( 1) k 1
 32 
 
1 5
48
k 1 ( k  2 )

8
1
=

  ( 1) k e 3( 2 k 1) log 2 
65
2 cosh log 8
k 1

4 .12310562561766054982... 
17
1 .12316448278630278663... 
3
1
erfi

2
3


2 .12321607812022000506... 
.123456789101112131415... 

1
 k !3 (2 k  1)
k 0
k
1   1 
 1 
  
  
 
4 2 
2
 2 
k (2k  1)


2k
k 1


 ( 2k
k 1
2k
2
 1) 2
Champernowne number, integers written in sequence


10
k
( 1) k k 4
  k  
81
2k
k 1 10
k 1
37  6
.12346308879239152315... 
  2 ( 3)
22680
2
1
1
=
 ( 6)   ( 2)  ( 4)   3 ( 2)   2 ( 2) 
3
3
3


1
(2 k )
.12360922914430632778...   2 k    k
k 1 3
k 1 9  1
.123456790123456790
J943
=
18 MI 4, p. 15

2
Hk

4
k 1 ( k  1)
.12370512629640090505... 
.12374349124688633191... 

1 .123912033326422...

8 .1240384046359603605...

.12414146221522759619... 
.12416465383603675513... 

 
=

 
=

529 .124242424242424242424 =
.12427001649629550601... 

 
=

 
=
.12429703178972643908... 
.12448261123279340605... 
.124511599540703343791... 
.12451690929714139346... 
1 .124558536969386481587... 
.1246189861593984036...


( 1) k 1


 1
2   ci    log 2  
k
 2


k  1 ( 2 k )! 4 ( k )

355
k
  ( 2 )  4 ( 3)  
3
108
k 1 ( k  4 )


1 
1 1   ( x)  dx 
66

( 1) k 1 Hk
3 24
log
  k
2
3
k 1 3 ( k  1)
 1  i 
i 
1  i 
 i 

  1     1      H    H     
2 4  2
4  2
 2 
 2 

1


2
k 1 2k ( 4k  1)

( 1) k 1  ( 2 k  1)


2 2 k 1
k 1
174611
  B20
330



coth   1 
1
 (i)   ( i) 
4
2 4
1
1  i  H 1  i   1  i  H 1  i   3
4



1
1
1
  3
 4

3
2
1
k 2 k  k  k  1
k 2 k  k
k 2 k  1
k 1

4
( 1) kHk
4
 1  log   
25 
5
4k
k 1

5
5
H
log 2   k k 
2
4
k 1 5 ( k  1)

(k )
( 1) k 1 k

8 1
k 1

1 
1

 log 2  



9 3
k  1 ( 6k  4)( 6k  1)
3
4
log 2 
11 log 2 
9
 (3) 
8
1
arcsin 3 x
 x dx
1
15 
k
 k
2 k 0 2 (k  1)(k  3)
J263
1 .124913711872468221...

8 log 2 

2 2
1
12   2
2
3
k 1 k ( 2k  1)
1
log 2 x
 1 1
   ,3,    4
dx
512  16 4  0 x  16
1
.12493815628119643158... 
( 1) k 1 k

k
k 1 2  1
2
2
( 1)1/ 4
( 1)1/ 4 
i
.12497829888022485654...  sinh
sin
  cos
 cosh

4
4
2
2
2 

sin k  / 2
 
= 

k
k 1 ( 2 k )! 4

.12497309494932548316... 

.12500000000000000000
=
=

1

8
k
 1) 2
k 1

Hk

k 1 ( k  1)( k  2)( k  3)
 (4 k
J364
2

=
=
k5

k
 (1) k
k 1 e

(k )


k
k 1 4  1
1

 
=
x
3
arcsin x arccos x dx
0

=

1  dx
2 5 
x
 log1  x
1
1 .125000000000000000000 =
4 .125000000000000000000 =
.1250494112773596116...
9
 H ( 3) 2
8

33
k3
  k
8
k 1 3
 1   ( 2 )  12 ( 4 )  6 ( 3)  8 (5) 
.125116312213159864814... 
.12519689192003630759... 
2
6

15

2
  ( 2) 
1
4




6 9 3 2

0
k3


5
k 1 ( k  2 )
(1   ( k ) )

k2
k 1
 ( 2)

 ( 4)
 /2



(1  sin x ) 2
dx
(2  sin x ) 2

1
1
 1
 (2)    
3
 8
8192
k  0 (16k  2)

( 1) k 1
.12521985470651613593...  
k
k  1 ( k  1)! 2  ( 2 k )
.12521403053199898469... 
1 .12522166627417648991... 


2 4
x4

 2 (2)   (4)   
dx 
4
3 45
0 sinh x
2 .125353851428293758868... 
6
4
9 (5)  2
17  24 (3)   2 (3) 2  9 (3)  97  5  1

2
96
4
22680 288 16

H k H k 2
k4
k 1
=


1
 Ei (2)  Ei (1) 
e

1 .1253860830832697192...
1
2 
n not squarefree n

1
e x dx
0 (1  x) 
Titchmarsh 14.32.1
2

2
Hk

k
6
k 1 2

1
1
(2)  1
.12546068941849408713...  
    
3
 6
3456
k  1 (12 k  10)
2 .12538708076642786114... 
 log 2 2 

csc 3
1
( 1) k 1

  2 2

6
18
k 1 k   9
3
5
1
7  3
.12549268916822826564... 
arctan

log 

6
3
18
3 12
1 .12547234373397543081... 

.12552511502545031168... 

 (k )
k 2
k
4


 1

2
dx
 x3
1
k 1


1    (k )  (k )  1  1  

3
4
k 2
=
x
 Li  k   k 

.1255356315950551333...

k 2 n2
1
  (  1) ,

 (k )
nk
where S is the set of all non-trivial integer powers
S
=
1

 a nontrivial
1


int eger power

3 .125552388163504646336... 
.12556728472287967689... 

 

 
3 (3)  2 1



2
12 2

HkHk
 k ( k  2) 
k 1
5 1
7 1 1 
1 
2 F1  2,2, ,  
2 F1  3,3, ,  
2 4
2 4 3 
2 
4 4 5
4 
1
1 5 
1 

arccsch2 
log

25 125
25 
2 
5
2

k 1 k
=  ( 1)
 2k 
k 1
 
k 
=
72
4
6


 9 ( 3)  3 (5)  28 
2
15 945


 (k )
1
 1
.12573215444196347523...   k 2   Li2 

 2k  2k
k 2 2 k
k 1
.12560247419757510714... 
2 .12594525836597494819... 
.12596596230599036339... 
.12600168081459039851... 
2
9
sinh  2 


2
 1  ( k  2)
k 1

1
1
 2
 ( 2)    
3
 9
1458
k 1 (9 k  7)

1
 1
Li4     k 4 
 8
k 1 8 k

2




k
k 1
6
1
( k  1) 3

1 
 12

 log 31  5 log
 1  2 5  2 5  
20 
6 5 

1
22
22 
 2 5  5 arctan 13 


10

2
5
arctan
13


 
10 
5
5 

dx
=  3
x  x 2
2
.12613687638920210969... 

 

 
.1262315670845302317...



5 2  4
1
 21

 5 (3)   (5)   5
3
2
30
k 1 k ( k  1)

7 (3)
1
1
 1
 (2)    
 
3
 4
512
128
1024
k  1 (8k  6)
 2
 
1
 3

1
1
3
.126321481706209036365...  
log 2
dx  GR 4.325.5
 
2 log log
x
1 x  x
 1
3
0
 
 3
.12629662103275089876... 
.1264492721085758196...

.12657458790630216782... 
3

1

log 2



2 3 3
3
1  3 (2)  4 (3) 
( 1) k


k  0 3k  5


x
1
6
dx
dx
 x3

 ( 1) k  ( k )   ( k  1) 
k
2
k 2
4k  k  2k  1
k 2 ( k  1) 3
k 1


 
=
11 .12666675920208825192... 
3

11

2

3 1
6 2


dx
 1 x
 7
 2
.12687268616381905856...  11  4 e
2
2 .12692801104297249644...  log1  e  


1
 ( 3k )
.1269531324505805969...   k    k
3
k 1 2
k 1 8  1
2
 4
 
 3

1 .12700904300677231423... 
4 i 4 i
     
 3 3  3 3

1
 1
.12702954097934859904...  Li3     k 3 
 8
k 1 8 k
1 .12673386731705664643... 
.1271613037212348273...

12 / 11
0
 
3 log 2 


4
8



1
 1  (3k  1)
k 1
1


2
k 1 16 k  4 k
2




( k )
k 2
4k


=
x
4
1
.12732395447351626862... 
3 .12733564569914135311... 
dx
 x  x2  x
3
2
5

2
2(2) 
1/ k 2
k 1
.127494719314267653131... 
 (2 k  1)

 2k k
k 1
.12755009587421398544... 
 ( 4 )  10 ( 2)  2 ( 3)  15 


k
k 1
4
13 (3)
4
1 ( 2 )  1
   

 
 3
216
432
1296 3
1/ 2
1/ 2

1
e e
1
cosh 
 
k
2
2
k  0 ( 2 k )! 4

1


1  2

2
 (2 k  1) 
k 0 
3
.12759750555417038785... 
1 .12762596520638078523... 
=
(1) k 1


k 1
1
k 1 2
(4)  1
.12764596870103231042... 
(2)  1

.1276340777968094125...

1 .1276546553915548012...

4 log 2 
1
=
1
( k  1) 3
2
1
 ( 6k  4)
k 1
3
AS 4.5.63
J1079
(1) k 1

k
k
k 1 2 2  1



6



1


3
2
k 1 2 k  k

( k  1)
k 1
2k


dx
 Li ( x ) x
2
2
2
0
.12769462267733193161... 
 1
2 arcsin 2   
 4
.12770640594149767080... 
1
5
log 
4
3

.12807218723612184102... 

 1 

( k  1)!( k  1)!

(2 k )!4 k
k 1


x dx
4
1
x
2
0 ( k ) 
k 1

2k 
1
2
3
 
.12812726991641121051...    

4
12 
4

28 .12820766517479018967... 
2

.12850549763598564086... 
k
1



 (x
0
2
dx
 1)( x  2)( x 2  3)
2
3 ( k )
2k

k 1 
2

  1 

k  2

k 1 
k 1
1 .1283791670955125739...
k
3

k 1
sin 1 ci (1)  cos1 si (1)  (1   ) sin 1
K ex. 123
1

 
=
  x log x sin(1  x ) dx
0
1 .12852792472431008541... 
(1)1 / 4

 i cot  (1)   cot  (1) 
4
1/ 4
3/4



k2
1
( 1) k 1  (4 k  2)  1




4
k

1
2
k 1
k 1


2 log 2
1
1
1
.12876478703996353961... 


  2
  k
3
3
k 1 6k  9 k  3
k 1 2 (3k  3)
 
=


 
=

1
.12878702990812596126... 
 5
 
 6
.12893682721187715867... 
2
1
1  dx
3 4
x
 log1  x

k 1
1
k ( k  2)
( 1) k
 0 F1 ;4;1 

k  0 k !( k  3)!

1 (2)  2 
1
.12895651449431342780... 
    
3
 5
250
k  0 (5k  2)

1
.12897955148524914959...  
k
k 1 (8 k )

 2 k  1 k
88
.12904071920074026864...  8 
  ( 1) k 1 


 k  16k
5 5
k 1
.1289432494744020511...

J3 (2) 

LY 6.117

1 1
log 2 x
 Li3 (4)   3
dx 
2 8
x  4x2
1

1
1
.12913986010995340567...  Li2     k 2
8
k 1 8 k

1
1 .12917677626041007740...   2 k 1
k 1 k

27
1
3
1
3
.12920899385988654843... 
log



(
)




5
3 
2
k 1 9 k  k

1
1 .12928728700635855478...   7
6
5
4
3
2
k 0 k  k  k  k  k  k  k  1
.129111538257100692534... 
.12931033647404047771... 


 







1
1  3 24  1  3   log1728  3 3 log 3  24 
4





3
33 3
1  3 log 64(2  3 )  log(2  3 ) 
2 3 
4
2
(1) k 1 (k  1)
= 
12k
k 1

2
( 1) k 1 k


 log 2  
2
12
k 1 ( k  1)


k 1
 ( k )   ( k  1)
= 



3
2
2k
k 1 4 k  2 k
k 2
1
1

log 2 x
x log2 x
x log x
= 
dx
3 dx  
3 dx   
( x  1)
( x  1)
( x  1) 2
0
1
0


 
.1293198528641679088...

 
J347

e1/ k

4
k 1 k
.12950217839233689628... 
e
.12958986973548106716... 
( 3) 
2
12


1

4
k
k 2
6
2k  1
 k5  k4  k3

=
 k  (2 k )   (2k  1)
k 2

2 .12970254898330641813... 
1
 ( k !)
k 0
3 .12988103563175856528... 
 HypPFQ {},{1,1},1
3
 1
 

.12988213255715796275... 
.12994946687227935132... 


1



 tanh   i   i      i  
2
2
1


3
2
k 2 k  k  k
1
 6


  (3k )   (3k  1)
k 1
CFG A28
arccos2 x
7 (3)
1 .13004260498008966987...   G 


 log 2  
dx
(1  x 2 )(1  x )
2
8
4
0
.13010096960687982790... 
1
2
2

1
7 ( 3)

4
2k  1

 (2 k  3)
3
8
k 0
3

Hk
1 .13020115950688002...
  k
k 1 3

1
1
k
.13027382637343684041... 

sinh  
k
4
2
k  0 ( 2 k )! 4
 5  5 
1  (1)  5  5 

  3  5  (1) 
.130287258904574144837... 
2 




2
2
1 5 






=
 (1)
k 1
.13030594417711116272... 
k
Lk k  (k  1)  1 
 
 2  2  1 
4 2




0
x(1  x)
dx
1  x8


1
 (k )  1
.13033070075390631148... 
log 2  1      2


2
k 2 k  k

k  1
1

 
=   1 
 ( k  1) log


k 
2k
k 2

1
2
.13039559891064138117...  10     3
3
k 1 k ( k  1)

 
( k )
1
.13039644878526240217...  
  j j

k  2 k ( k  1)
k  2 j  2 k ( k  1)

1 .13039644878526240217... 
  ( k )  1 ( k )  1
0
k 2


 
1
=   j

k 2 j 2 k  1
 
( k )
m
m 1 k  2 k



=


K ex. 127


   (mk  m)  1
k 1 m  2

   ( jk )  1 
j  2 k 1
( k )  1


k  2 k ( k  1)


( k )
 k 1 
k 2



2 .13039644878526240217... 
  ( k ) ( k )  1
k 2
.13039943558343319807... 

0
2
log 2



4 32
2

 
=
2 .13068123163714450692... 
.13078948972335204627... 

 
=
2
x dx 
GR 3.839.1
0

k
2
8

4 log 2
2


(1  2 ) 
log(1  2 ) 
49 28
7
7

1
 
= 
Prud. 5.1.7.24
k  1 k (8k  7 )

1 
( 1) k
.13086676482635094575... 
 csch 2    2
8 2
k 0 k  4
  3  2i 3 


1

 cos   3  2i 3  cosh  3 
1 .131074170771424458336... 
cos




3
2
2
2




.13086510517299719078... 

 x tan
log k
(1   ( k )) log k

 
2 2
(1  k )
k2
k 2
k 2
 
j log k


k2j
j 1 k  2

3e
1
k 1

 
4
4e
k  0 ( 2 k )!
1
3 log 2  3 log   2   12 (1)  1 
6


 1
1
 1 
k  (k )  1



  k  k 2 log 1   
(
1
)


2
k2
 k 
k 2
k  2  3k

1 .13046409806996953523... 
 /4



 
=
 1  k
3

1

3
(k  1) 

cos 1/ 4  cosh 1/ 4

1 .13114454049046657278... 
2
k 1
.131263476981215714927... 
1 .131279000668548355421... 
.13129209787766019855... 

2
4

k
 (4 k )!
k 0
1
 4G  6    arcsin 2 x log x dx
0
 i ( 1)  cos ( 1)1/ 4    i ( 1)1/ 4  cos ( 1) 3/ 4  
3/ 4
1
 
7
 
1
8



 
=
1 .13133829660062637151... 
.131447068412064828263... 

 
=
.131474973783080624966... 
.13150733113734147474... 
4 .1315925305995249344...

.13178753240877183809... 
Hk
k 2
k
k 1

1

k
k 1 k k !
1
  ( 2 ) (2  i )  ( 2) (2  i )   (3, 2  i )   (3, 2  i ) 
2

x 2 cos x
0 e x (e x  1) dx

7 ( 3)
1
 
3
64
k 1 ( 4 k  3)

( 1) k
3
 ( k )  1 


k
k 2

1

k  2 2  ( k )
8

k 1

.13179532375909606051...   4
k  2 k log k


log 7  3
3
5
arctan



6
6
3
3



x
2
 1
1 
 4
 3
 

k log k 
k  2  k log k

2
 4G 


 
=
 1   ( k ) 
k 2
1/ 2

dx
1
3
1  ( 3)  3 
( 3)  1  
        
 4
 4
4
384 
k

 log 2
( 1)
.13197175367742096432... 

1  

4
2
k  0 ( 2 k  2 )( 2 k  3)

1
 
= 
2
k 1 8k  6k  1
.131928586401657639217... 
 
=
 7 
 
8
 (3)  Li3    Li3    log  log log 8  2 Li2    
8
8
2
7
7
8

1  dx

2
0 log(1  4 x ) dx  1 log1  x 4  x3
4
  ( s)  1) ds
3

( 1) k 1 k 2
 ( k  1 / 2)
k 1
4
1 .13197175367742096432... 
=
=


log 2
( 1) k 1
 
4
2
k 1 2 k ( 2 k  1)

1

2
k 1 8k  10k  3

(1) k / 2 

k 1
k 1

1
=

Prud. 5.1.2.5
1
4  dx

 x log1  x
0

=
J154


1
 log1  2 x( x  1)  dx
0

=
.13197325257243247...

3 .13203378002080632299... 

.1321188648648523596...

.1321205588285576784...

.13212915413764997511... 
.13223624503284413822... 
.13225425564190408112... 
.1323184679680565842...

11 .13235425497340275579... 
.13242435197214638289... 
.13250071473061817597... 
25 .13274122871834590770... 
arctan x
dx
2
x
1


ie i
sin k
i
2i
i
Li3 (e )  e Li3 (e )  


3
2
k  1 ( k  1)
3 log 3
  ( 13 )

 1

GR 8.336.3
 

     
 3
2
 ( 13 )
2 3

2
2
k 1 kH k

1  log    (1)
9
3
2k
k 1

1 1
( 1) k 1
 
2 e k 1 ( k  2)!

6
7
( 1) k 1 Hk
log  
7
6
6k
k 1

1


4
k 2 k  7

 ( 5k )
 k
2
k 1

5
1
cos3 k
(i   2 )  Li3 (e3i )  3Li3 (ei )    (1) k 1
8
4
k3
k 1

5815 3
k6
e  
k
729
k 0 k !3
log(   2 )
 3
2 log 2
1



21
7
14
8

1 .132843018043786287416... 
 ( 1)
k 1
k 1
(k )
2k  1

( 1) k

k  0 ( k  3)( 3k  2 )


1
1


( 1) k e 2( 2 k 1)

2
2
2 cosh 2
e e
k 0
9
1 .133003096319346347478...   
 4

1
 2 3 log 2 2
.13301594051492813654...  2 log 2 

3   k 2
2
4
2
k 1 2 k ( k  1)
  

5 
2
3 
.13302701266008896243...  1 
 cot  cot  
 log sin  log sin 
16 
8
8
4 
8
8
.13290111441703984606... 
( 1) k


k 0 4k  5

=
1 .13309003545679845241... 
.13313701469403142513... 
1 .13314845306682631683... 

x
1


4 log 2
2

0
  8
x
6
dx
 x2
dx
 2 x
(1)
8
e 


1


k
k  0 k !8


1
 k !! 2
k 0
k
1
.13314972493191508633... 
1
12   2    x log x arctan h x dx 
16
0
.13316890350812216272... 
1



csch

2 2 5
5
.1332526249619079565...
1 .1334789151328136608...


J943
( 1) k 1

2
k 1 5k  1


2
2  2 log 2  log 2 2 
3(5)  (2)(3) 
k 1
Hk

k
k 1
4
k
Hk
( k  1)( k  2 )


 2 k  ( 1) k 1
5
2
.13348855233523269955... 

 log 2  log 1  2     k

12
3
k  1  k  4 k ( k  2)



1 .13350690717842253363... 
  (2k ) (2k  1)  1
k 1

1
 1
log 8  log 7  Li1     k
 8
k 1 8 k

1
1
= 2 arctanh
 2  2 k 1
15
( 2 k  1)
k 1 15
k 1

( 1) kHk
3
3
.13355961141529107611... 

 1  log   
16 
4
3k
k 1
.13353139262452262315... 
.13356187812884380949... 
 (3)
9

6 .13357217028888628968... 
log(1  x 3 ) log x
 
dx
x
0
1
 2k  1
  k  k ! k
k 1
K148


1 
1 1 
1
 (2 k  1)
.13360524590502163581...       1      1    

2   2  
 2k
k 1

1
 
= 

2 2
k 1 k (  k  1)

8
4
(1) k 1 k
 (k  1)  1
.133790899699404867961... 
   log
 
9
3
k 1 ( k  1)( k  2)
1 .13379961485058323867... 
7
2
arcsin

2
7
 2k 

  k  7
k 0
k
1

(2 k  1)

.1338487269990037604...

 .1339745962155613532...

.13429612527404824389... 
 
1
sin 2 x dx
0 e x 

( 1) k 1 ( 2 k  1)!!
3
 
( 2 k )! 3k
2
k 1

1
k
log

2
k 1
k  2 ( k  1)
1
2 3 coth  csch 2  3 2 csch 2  3 coth   8 

16

1

2
3
k 1 ( k  1)

coth 2 1
e4  3
1
 



4
2 2
4
8
8( e  1)
k 1 k   4
 1 
.13402867288711745264... 

2
cos 2  2 sin 2  5


5
10e
=
.13432868018188702397... 
.134355508461793914859... 
2

27 3

x
3
0
1
.13451799537444075968...  arctan 2 
e
.13458167809871830285... 
3
dx
 27

e
x
2
dx
 e x
k 1
( 1)
k
2
k 1
1
1 .13459265710651098406... 

arcsinh 1

J951

1
1 .13476822975405664508...   k k 1 
k 1 k 2

arcsinh 2 1
1 


 2k 2 
k 1 

1
x
 x /2
dx
Prud. 2.3.18.1
0

2 .13493355566839176637... 
.13496703342411321824... 
=
e
e x
 
dx
4
1  x4
0
2

11
1

  2
2
12 16
k 1 ( k  1)


1
1



2
2
4
2 2
)
k 1 k ( k  2 )
k  2 k (1  k

J400

=
 ( k  1) (2 k )  1
k 2

Hk 2
1 .13500636856055755855...  
k 1  2 k 
 
k

cos1
( 1) k k 2
 
4
k 1 ( 2 k  1)!

3
(1) k 1 H E k
1

.13515503603605479399... 
log  
3
2
2k
k 1
.13507557646703492935... 

.13529241631288141552... 
20 .13532399155553168391... 

.1353352832366126919...

 

 
1 1
  Li  3   3  
k 1

k

2
49
x log x
dx

 
3
9
x
0 1
Ai (1)
1
2
1 .1352918229484613014...


2 cosh 3  e 3  e 3 


1
( 1) k 2 k
( 1) k 1 2 k ( k  1)  ( 1) k 2 k ( k  3)






e2 k 0 k !
k!
k!
k 0
k 0

( 1) k 1 k 2 k
= 

( k  2)!
k 1


=
log 3 x
dx
3
x
e
(

)
0


1 .1353359783187...

k 2
.13550747569970917394... 
 
=
4
(k )

2 cosh 


(k )



e  e 


 (1)
k 1
22 k 1  (2k  1) 
k 1
cos 2 x
 cosh x dx 
GR 2.981.3
0
.135511632809070287634... 
1 .13563527673789986838... 
.13574766976703828118... 
1  
1
 1 1 
  
 
  1 
 2 4  
4  
4 
9
e x dx
  1  e 2x 
0

I 2 (1)

cosh 11 1
1
1
1
 1  dx
 sinh 2  sinh 1 tanh   sinh  4  5
4
2
2
4
2
 x x
1

k!
.13579007871686364792...  
k  0 k ! 1

163
1
.13583333333333333333 =
 
1200
k 1 k ( k  1)( k  5)
.13577015870381094462... 

5 .13583989702753762283... 
 k  ( k )  1
3
2

k 1
(1) k 1 2k k
 (k  1)  1

k 1 ( k  1)( k  2)

k2
.13593397770140961534...  11 e  18  
k
k 1 ( k  2 )! 2
.13587558784724491212... 
5
1
 log

2
6 

  1  i   1  i  
2 sinh 


(1) k
(1) k 1
 2
=  2

k 2 k  1
k 1 k  2k  2
.13601452749106658148... 

 

 

1
=
cos(log x )
dx 
(1  x ) 2
0

GR 3.883.1


 
=


=
cos(log x )
dx 
(1  x ) 2
1


sin x
dx 
x
 1)
 e e
x
0



16 23
( 1) k

 
e
4
k 1 k !( k  4 )


 (k )
 1
1 .13607213441296927732...   2   Li2  
k
k 2 k
k 2
.13607105874307714553... 
 1
3 .13607554308348927218...  2 ( 4)  2 Li4    
 2
.13608276348795433879... 
.13610110214420788621... 

 
=
1 .13610166675096593629... 
1 .136257878761295049973... 
.13629436111989061883... 
1
log 3 x
0 ( x  2)( x  1) dx 
( 1) k k  2 k 
 
 
8k  k 
3 6
k 0


1
 ( 3k )
( 1) k 1 k 



3
8
k 1 8 k  1
k 1
3i 3
3i 3
2 3
3  3i
 1 log 2



 

 

6 3
3
6( 3  3i )  4  6( 3  3i )  4 

1
4 e1/ 4  1  
k
k  1 ( k  1)! 4


k !k !
16
16
1
1
arcsin  



k
15
4
15 15
k  0 ( 2k )!4
k  0 k  2k 
4  
k 

5  ( 1) k
( 1) k
2 log 2    3
 GR 1.513.7

4 k  2 k  k k  0 ( k  1)( k  2)( k  3)
1


=
k
k 2
.13642582141524984691... 
 1 1
 Lik    
k 1
  2 2

1  
1
(1) k  (k )

 1      2 2
 
k
    k 1 k   k
k 2
1
 2 (k  1)
1

 

 (1)

k 1

2 .13649393614881149501... 
  ( k ) ( k  1)  1
1
k 1
.13661977236758134308... 
1
 
 2

=
 ( 1)
.13678737055089295255... 

 
=
1 .13682347004754040317... 
.13685910370427359482... 
.13695378264465721768... 
.13706676420458308572... 
.13707783890401886971... 
sin  3

 3

2
 
=

72


4k  1
 (2 k  1)!!


 (2 k )!!  (2 k  1)(2 k  2)

13 (3)
2 3
1 (2)  2 
1

      
3
 3
27
54
81 3
k  0 ( 3k  2)

5
 2 5
k2 1
coth 

  4

2
4
2
6
2
k 1 4 k  k

( 1) k
 (2 k )   (2 k  2)

k
k 1 4


1
k3


( 1) k 1  (5k  3)  1


5

1
2
k
k 1
k 1
2

1

( 3) 
6   2
3
2
k 1 k ( k  1)

3
1

1  3 log   k
4
k 1 4 k ( k  1)



k 2
1
 (6k  3)
1
  (12k  3)
k 1

 1  k
k 1

2

2
 

 
.13711720445599746947... 
14 .13716694115406957308... 

 (3)   (k ) log k

k3
 2 (3) k 1
9

2

2 

 1
1

2 
(12 k  9) 
1
k 1  2k  2
4  k
k 

1  dx

=  log 1  6 

x  x
1
=
2



J385
3
k 1
k 1
.13675623268328324608... 
2
1
 (2 k  1)!!




k  1  ( 2 k )!!  2 k  2

2
J391
J149
1 .137185713031541409889... 
.13737215566266794947... 
.13744801974139900405... 
.13756632099228274...

.13786028238589160388... 
2 .13791866423119022685... 
.13793103448275862069... 
1 .13793789723437361776... 
12 .13818191912955082028... 
2
log 2 2  2 log 2
7 (3)

 8 log 2 

2
2
2
2
 ( 3)   (5)
( 3)

2 
log 2
1


 
9 12
2
k  1 k (8k  6)
k

( 1)

3
k 2 k  2

1
  e 2( 2 k 1) 
2 sinh 2
k 0

4
2
2
4
x dx

 
csc
5 5 5
5
5
1  x5/ 2
0

4

29

Hk
 k (2 k  1)
k 1
2
AS 4.5.62, J942
(1) k 1 F2 k

4k
k 1


e1/ 4 

2
e
x2
cosh x dx 
0

  log1  x  dx
 2 1 3 

12
0
.13822531568397836679... 
log  1  e 1  
ek
k 1
1 .13830174381437745969... 

1 
 1

     log 1   
 ek  
e
k  1  ek
( k )

=


k
I2 2 e


1
F ;3, e 
20 1
e
1

k 1
k 1 k
5 3
 (3)


36
81 3

ek

k  0 k !( k  2)!

1 .13838999497166186097... 
1 .13842023421981077330... 
.1384389944122896311...

2 log 2 2 

 ( 1)

1
1
1



3 
3 
3
 (3k  1)
(3k  2)
(3k  3) 

4  dk
k
  K ( k )  log k 
0
9   3  9 log 3

4
log 2
.13862943611198906188... 
5
1 .13847187367241658231... 
k
k 0
1
2
12


( k  13 )!
2 )! k
3
 (k 
k 1

.13867292839014782042... 
4 log 2  2 log(2  3 ) 
1  2k 

k 
k k 
 16
k 1
GR 8.145

=
k 1
1 .13878400779449272069... 
.1388400918174489452...

( 2 k  1)!!
k
k
 (2 k )! 4
( 1) k (12 k 2  1)
k ( 4 k 2  1) 2
k 1


dx
  4
x  16
16 2
0
2 G  log 2 



( 1) k 1 k
1



5k
k 1 k ( k  2 )( k  3)
k 1

k2
= 
k 1 ( k  1)( k  2 )( k  3)( k  4 )

e  pf (k )
2 .13912111551783417702...  2 e  2 e  
2k
k 0


4
1
20k 2  5
(1) k 1 k!k!
.13918211807199192222...  1 
arcsinh  
 
2
5k
5
k 1
k 1 ( 2k  1)!
.138888888888888888888 =
.1391999869582814821...

5

36



2arcsinh 1  2 log 1  2  2 log 2  2 
=
( 1) k 1 ( 2 k  1)!!

k 1 ( 2 k )! k ( 2 k  1)


 
.13920767329150225896... 
.13929766616936680005... 
.13940279264033098825... 
1

3

csch 

6 2 6
2

2
1  erf (1)

e
5

1 
4 4e
.13943035409132289...

     (3) 
5
 2
4
 /4

0
x 2 tan 2 x
dx 
cos2 x
( 1) k 1

2
k 1 2 k  3



e
 x2
dx
1
.1394197585790642108...


1
 1  dx
 1  dx
1 sinh x 2  x 7  2 1 sinh x  x 4
3 1

F1  2,2, ,  
2 4

9 3
2



(k!)
2k
k
 
 
= 
 
k 1 ( 2k  1)!
k 1  2 k  1
k 1  2k 


 
 k 
k 

1
E2 k
1 .13949392732454912231...  sec
  ( 1) k
2
( 2 k )! 4 k
k 0

log 2 log 3
H
.13949803613097262886... 

  2 kk  2
3
12
k 2 4
2 .13946638410409682249... 


1   log 2   2
1 
 
 G 
3
4
2 16


4

3
2
GR 3.839.4
3 .13972346501305816133... 
log e  e   
1 cos1  2 sin 1
sin x dx



2
5
5e
e2 x
0
7
=

50

1
=
  ( 1) k e  ( 2 k 1) log 7 
2 cosh log 7
k 0
1
.1398233212546408378...
.14000000000000000000

 



 
=
 ( 1)
k 2


 
=
x
1
1 .1405189944514195213...

5
23 .14069263277926900573... 
18 .140717361492126458627... 
.14095448355614340534... 

2

1
3

log(2  3 ) 
2
3


1
 3 (2k  1)
k 0
k
k
9
k 1
k
sinh  
Shown transcendental by Gelfond in 1934

 k
2k
 (2k )! 
k 1
1 
3
 1 
 8 24 G  2 2 log 2  9 (3)    ( 3)     ( 3)    
2048 
4
 4 

0
2 .14106616355751237475... 
2k
e   i 2i 
 /4
=
( k )  1
3 arctanh
=

k
dx
 x4
9

64
2 .14064147795560999654...   ( 9 ) 
.14062500000000000000

 /2

1
2 4 log 2  18 2 (3)  93 (5)   x 3 log sin x dx
128
0
1
.14002478837788934398... 
4

1
.14004960899154477194..   2 k

Berndt 6.14
1
k 1 e


5
1
(1) k
 log 2  
 
.14018615277338802392... 

6
k 1 2 k ( 2k  1)
k 0 k  4

1
 
= 
2
k 1 4k  10k  6
.14002410170685231710... 

J943
x3
dx
tan x
e  

.14111423493159236387... 
1
2 2
k
k 1
.14114512672306050551... 
2k



cos3 k
1
i 3
 3i
i 3
3i
log 1  e  1  e  1  e 1  e    
k
8
k 1
  2 2
 Li2     

k k
k 2 

1 .14135809459005578983... 
2 k  ( k )  1

k2
k 2


7 .1414284285428499980...
10 .141434622207162551...
51

2 2
 1
 2 log 2 2  4 log 3  4 Li2    
 2
3

. 14147106052612918735... 
2 .14148606390327757201... 
4

9
1
log 2 x
0 ( x  12)3 dx 
2
 2k 
k
 


k
k  0  k  16 ( k  1)( k  2)

3log 2  2 log 3 

3
( 1) k 1 / k
k 1

( 1) k

k
k  0 2 ( k  5)

k sin 6 k
.1415926535798932384...    3   ( 1)
k
k 1

k sin 4 k
1 .1415926535798932384...    2   ( 1)

k
k 1
.14155012612792688996... 
32 log
3 77


2
6
1

 
=
 arccos
2
x dx 
0
 /2

 
x
=
2
sin x dx 
0
3 .1415926535798932384...




 
 


=
 , 
=
 1 
 1 
 1 
 1 
48 arctan   128 arctan   20 arctan
  48 arctan

 49 
 57 
 239 
 110443 


=
 1 
 1 
 1 
 1 
176 arctan   28 arctan
  48 arctan
  96 arctan

 57 
 239 
 682 
 12943 


 1 1
 2 2
Borwein-Devlin p. 74
Borwein-Devlin p. 74
50k  6

k  0 k  3k 
2  
k

=

Borwein-Devlin p. 94



3k  1
8



(
k
)

1


k
k 1 4
k 1 ( 4 k  1)( 4 k  3)


dx
e x / 2 dx
  x
= 

1  x2
e 1




 

 

 
=

=
Vardi, p. 158
x
 1  sin x dx 
0


 
=
dx

2  cos 2 x

0


 
=
 log1  x  2  dx 
0
 
=

 log1  x
0



 log(1  x
2
)
0
2
3 
 dx 
 1
dx
x2
GR 4.295.3
logx 2  (e  1) 2 
dx 
0
x2  1

=
 1  x2 
=  log 
 1  x 2 dx 
x


0
1

 
2
GR 4.298.20

=
1

  log x log1  2 dx
x 

0

=
log( x 2  (e  1) 2 )
dx 
0
x2  1
1
=
 arccsc x log
2
x dx
0
3 1

6 .1415926535798932384...    3  2 F1  2,2, ,  

2 2
7 .1415926535798932384...
12 .14169600000000000000

 4 
= 12
2 k 1

k 1  2k 
 
k 

2214

15625

k6


k
k 1 6

2k k


k 1  2 k 
 
k
1
.141749006226296033507... 
  log 2 (1  x ) log 2 x dx
0
log 2  2
  log 2

 

2
32 4
8
 /4


=    x tan x tan x dx


4
0
.14179882570451706824... 

 

Bk  k


e  1
k!
k 0

1
( 1) k
.14189705460416392281...  arctan
  2 k 1

7
( 2 k  1)
k 0 7
.14189224816475113638... 

 
 (1)
=




k 1
k 1


 

=
 2 

arctan
2 
 ( k  2) 
[Ramanujan] Berndt Ch. 2, Eq. 7.5

1
 arctan 2( k  3)
k 1
 

1 .142001361603259308316... 
.142023809668304780503... 
.1421792525356509979...

1 .14239732857810663217... 
.142514197935710915709... 
1 .14267405371701917447... 
.14269908169872415481... 
.1428050537203828853...




[Ramanujan] Berndt Ch. 2, Eq. 7.6

7 (3)  2
k ( k  1)( ( k )  1)


8  
2
2
2 k 1
k 2

(k )
( 1) k 1 k

7 1
k 1

(2k  1)!!
1  3
4      1    2 log 2  
 4  4
k 1 ( 2k )!k ( 4k  1)
4

11
 2 log 3 2 log 4 2
 1  21 (3) log 2
6 (4) 

 6 Li4   
 2
4
4
4
log 3 (1  x )
0 x dx 

.14260686462899218614... 
2

1
=
J152



GR 3.797.1


8
2k
k 1
2
25

 (k )  (k )
csc

2

5
1

4
log 3 x
1 x  1 dx 




2
1
  (5k  1)
k 1

 (x
1
2
2

1

2
(5k  4) 
dx

 1) 2
2
14
2

log 
3
81 27

 (1)k
k 1
Hkk3
2k
( 1) k  ( k )  1

.14280817593690705451...  
k2
k 2
1
.142857142857142857
=
7

1 .14327171004415288088... 
3 .1434583548180913141...

.14354757722361027859... 
1

1
  k  Li   k  
2
k 2

3G
log e  e   
3
216

.14359649906328252459... 


 (k )  1
k 1
( k )
3 .14376428513040493119....  
k  2 Fk
k 3

1
 1 1 
    2  log 1   

k
k 
k 2  k


1 ei
cos k
ei
1 .143835643791640325907...  e
cos(sin 1) 
e e
 

2
k 0 k !
= cos(sin 1)cosh(cos1)  sinh(cos1)  

1
4
1
1
.14384103622589046372... 
log  arctanh   2 k 1

( 2 k  1)
2
3
7
k 0 7


dx
dx
=  3
 
x x
( x  1)( x  2)( x  3)
2
0

cos 1

.14385069144662706314... 
5

2e 4


cos 4 x
2 2 dx 
)
 (1  x

.143954174750746726243... 
 







5
2 
5 5 
10 3  5 log  5  5 log 2 

  25  11 5 log

4
8 
53 5 
5


1



 
=
4
k 1
1 .14407812827660760234... 
1 .14411512680284093362... 
.14417037552999332259... 
Fk F k
k (k  1)
 5
Li2  
 6
3 log 2  2G 


2
2
4
( k  1) ( k  1)

4k
k 1

1
 1
Li4     k 4
 7
k 1 7 k

=
k
8k  1
2
k 1 k ( 4k  1)


GR 1.449.1
K146
.14426354954966209729... 
.14430391622538321515... 
2 log



4
( 1) k 1 k

k
k 1 2 ( k  1)
2
dx
 e x
x
3 2


2 3
 e
 x4
0


GR 3.469.3

  e  x x log x dx
2
=
0

3 .14439095907643773669... 
 
k 2
2
( k )  1
2
.14448481337205342354... 
e e 1/ e  1  e 1/ e 
.14449574963534399955... 
 3 (2 k  1)

k 1
( 1) k 1 k

k
k  1 ( k  1)! e

H ( 3) k
k

2k

4
k 1 k ! k
2
 3  Li3 (3)  
.144713210092427431599... 
9
2 .14457270072683366581... 
.144729885849400174143... 

 1  log   log


log 2 x dx
1 x3  3x 2
 ( 2k )



4 ( 2k 2  k )

k  1/ 2 
k 1
 log
1 .144729885849400174143...  log    (1) 

k 
 2k
k  
e
k 1
k

Prud. 5.5.1.17
k 0

1
 3
31 (5)  32    5,   
1 5
 2
k 1 ( k  2 )


1
1
( 1) k 1  ( 2 k )
 
1 .14479174504811776643...   2 cos
k
( 2 k  2 )!
k 1 k
k 1
8

4 log 2
2
.14490138006499325868... 
 ( 2  1) 

log(1  2 ) 
25 10
5
5

1
 
= 
k  1 k (8k  5)
.14476040944446771627... 

.14493406684822643647... 

2
6
 

 
3

2

1

3
2 
k 2 k  k

 ( 1)  ( k  2)  1 
k 1
k 1

Hk
(2 k )  (2 k  2)




2
4k
k 2 k  k
k 1

4x
dx 
= 
2 2
(1  x ) (e 2 x  1)
0



=
Prud. 5.1.7.23
1

 
=
 log(1  x
2
)
0
1 .14493406684822643647... 
2 .14502939711102560008... 
2
6

1

2

dx

x
GR 4.29.1

 ( 1)  ( k )   ( k  1)  2
k
k 2
2/3
(1) k  1 k 7

2k
k 1

.14540466392318244170... 
381

2187
.14541345786885905697... 
2  log( e  1) 
2

1
 ke
k 1
2k

GR 1.513.3
1 (1)  3
8 2 5
1
 7
2



  ( 1)   
   8 25
 5
 5
25
5
25

1
 
= 
2
k  1 (5k  2 )

1
 1
1

.14552316993048935367...  Li3     , 3, 0   k 3

 7
7
k 1 7 k
.14544838683609430704... 

1 .145624268262139279409... 
2
3

12
32


 
=

2
11
1
3 log 5  8 arctan 2  44  

25 100
 (k )

.14581354982799554765... 
 3G 1 3 2 
 2   7 (3)
 
 
 log 2  G    

3
4
4
18  2
3 16 
Hk
 (4 k  3)
k 1
.14573634739282131233... 


k 2
.145833333333333333333 =

 k !( k  1)!   
k 1
7

48

1


2
k 1 k  4 k

 xe
x
log x cos 2 x dx
0
1
 2 
k I1 

 1
 k
2k 

1


k  0 ( k  1)( k  5)
( 1) k 1


2
k 3 k  4

1

 
=
x
3
log( x  1) dx 
0


 
=

1  dx
 log1  x  x
5
1
53
 log   56 (6)  140 (4)  56 (2) 
24

 ( 2k )  1
= 
k 4
k 1
.14583694321462489756... 

 
.14585774315725853880... 
log 2 1 2G  1
 

2
4
2
.14588643502776689782... 
(1) k 1 k 2
 (2k  1)

2k
k 1
2
1
 (2 k  2)!!




(2 k )!!  2 k  2
k 2 

J385

73 5
  4
2
log 2 2
.14606785416078990650...  2 log 2 
1 
2
2  3
3 .1462643699419723423... 
.14589803375031545539... 
.1463045386077697265...

2
4
4
 2 log 2 
( 1) k 1 Hk


k 2
k 1


1
 k (2 k  1)
k 1
2

GR 0.236.7
1
=
 log(1  x
2
) log x dx
0
.14630461084237077693... 
1
3
 log  2 
2
2
2 2 arctan
( 1) k 1

k
k 1 k ( 2 k  1) 2


1
8
k 2

cos x
dx
.14641197161688952006...   x / 2
e 1
0
.14632994687169923278... 
.146443311828333259614... 
.1464466094067262378...

.1464539518270309601...

k
4

 cosh  sinh   csch 
4
2

(1) k 1 k 2

2
2
k  1 ( k  1)

1
1
2 2


2 2 2
4
e
1 2 cos1  sin 1
log x sin log x
dx 

 
2
2e
x2
1
1

 
=
 xe
x
sin x dx 
0

1 .14649907252864280790... 
.146581405847371179872... 
2 .146592370069285194862... 
.14665032755625354074... 
1
1
1
PFQ {1,1,1},{2,2,2},1  
  e x log 2 x dx 
2
20
k 1 k!k
  1  log
9

16
6
 2  
5
2
(1) k 1
 (k  1)  1 

k
k 1 2 ( k  1)

cos x
 3 / 2  cos x dx
0
24  13 cos1  20 sin 1 
( 1) k 1 k

k 1 ( 2 k )!( k  2 )


1 .14673299472862373219... 
e
k 1
.146997759396759645699... 
2 .147043391355541844907... 
1
k /2
1

Berndt 6.14.1

 3
  
4 2  2 
1

3

3

8
x dx
1
e 
2 x
0
(1) k  1 k 4

2
3
k 1 ( k  1 / 4 )

32
i (1)
i
.14711677137965943279... 
 (2  i)   (1) (2  i)   (2, 2  i )   (2, 2  i )  
4
4


k
k 1
=  (1) k  (2k  1)  1  

2
2
k 1
k 2 k  1






 
=
1
x sin x
dx
x

2 0 e (e x  1)
(1) k  1
  k
k 1 2  2

.1471503722317380396...
.14718776495032468057... 
.14722067695924125830... 
=
9 log 3 1


40
10

(sin x  x cos x)3
dx
0
x6
Prud. 5.2.29.24
log 2 2  2
 1


 log 2 log 3  Li2   
 2
2
12
1  1
log 2 log 3  log2 2  Li2  
2  4
log(1  x )
dx
x2
0
90
1

4
  90  ( 4)  1
1
=
12 .147239059010648715201... 
.14726215563702155805... 

3

64
.14729145183287003209...  1 
1
x
3
arccos xdx
0
Ei (1)  1  

e

 ( 1)
k
k 1
Hk

( k  1)!
1

 
=
  e x 1 x log x dx
0

.14747016658015036982... 
 ( 1) k 1 k 2  (3k )  1 
k 1
.14758361765043327418... 
k 3 ( k 3  1)

3
3
k  2 ( k  1)

1
arctan  

 2
1
Plouffe’s constant, proved transcendental by Barbara Margolius

.14766767192387686404... 
 9k

3
1



 ( 3k )
9k

8
(1) k
.14775472229893261117...  5 
  k

e
k  0 2 ( k  3)!
2 arcsinh 1

 1
 .14779357469631903702... 
2
2

( 1) k 1 ( 2 k  1)!!
 
= 
2
k 1 ( 2 k )! ( 4 k  1)
k 1

1
k 1


2 arcsinh1
2 1


 log 1  2 
2
2
2
2
1 .14779357469631903702... 
1

 
=

1  x 2 dx
0
(1) k log k


k2 1
k 2

3
1
log 3  1   3


2
k 1 9 k  k

 ( 2 k  1)

9k
k 1

k
1/ 4
4  3e  
k
k  0 ( k  1)! 4
H5 / 2 ( 3 ) 

.147822241805417...

.14791843300216453709... 
=
.14792374993677554778... 
1 .14795487733873058...

.14797291388012184113... 
1  tan  2


7
8 2

( k )
1 .148135649546457344478...   k
k 1 2 k
1 .14812009937000665077... 
.148148148148148148148 =
4

27

3 .14821354898637004864... 

2
1
coth 2  
4
4

k 0
k 
k 1
2

 4k
k 1
2
J375
1
2
2
1
 4k  7
(1) k ( k  1)

2k
k 0

 (k  2)
( k !)
2


1
k
k 1
2
 1
I0  2 
 k
1
1
  1
Li2      ,2,0    k 2
7
7
 k 1 7 k

1
 1  10
.14834567213507945656...  4 2 arctan




k
 2
3
k  0 2 ( 2 k  5)
.14831179749879259272... 
1 .14838061778888222872... 
 
27
 3
log 3
1  1 1 
1
 hg     2

18
2
3  3
3 k  1 3k  k

(1) k  ( k )
= 

3k
k 2
.14839396162690884587... 

3
1


 
=
x
4
1

dx
 x3  x2
1
3
 2 
4 4e
1/ 5
1 .1486983549970350068...  2
.14849853757254048108... 
( 1) k 2 k

k  0 k !( k  2 )


1 .14874831559185321424... 
1
k
k 0
.14885277443216080376... 
.14900914406163310702...
.14901768840433479264...
8 .14912752141674121819...
.14918597297347943780...
 2 k  2

k 
 16 (k  1) 
16 ( 7  4 3 ) 

 1
4 Li3    2 
 2
log 2 x
1 2 x 3  x 2 dx 

( 1) k 1 (2 k  1)!!
1 1 
 2 log 
 
2
(2 k )! 3k k
3
k 1

log k
  3
2
k 2 k  k

4 e3  7
3k k 2


9
k  1 ( k  2 )!

3 7
(1) k
  k
 9 log 

2 2
k  0 2 ( k  1)( k  2 )( k  3)

1
1
=   k
2 k  0 3 ( k  2)
.14919664819825141009... 

8e
9/4
e
2

 1 
e
 x2
sin 2 x cos x dx
0

k
3
3 k
  (1) k 1
3
e k 1
k!
3 1

6 .14968813538870263774...  2 2 F1  2,3, ,  
2 4

.14936120510359182894... 
.149762131525267452517... 
19 
  3 log 2 
5
2

1 .14997961547450537807... 
 2  (k )  1
k
k 2
3

k!(k  1)!
 (2k  1)!
k 1

1


k  2 k ( 4 k  1)
(1) k ( ( k )  1)

4 k 1
k 2

.15000000000000000000
=
3
20
 (k )  1

.150076728375356585727... 
 (1)k
2 (k  1)
k
k 2
.15017325550213874759... 
.150257112894949285675... 
2
 3
 (3)
8

sin
 3
2
 
=

 
=

1
1 
 2k log1  2k 
k 2


3 
2

  1  4k

k 1
(1) k 1 H k


2
k 1 ( k  1)


1

3
k 1 ( 2k )

1


1
log 2 (1  x 2 )
log(1  x) log x
dx   
dx 
0
x
x
1

0
1 1
1 .15050842410886528915... 
=
log(1  xy )
dx dy
1  xy
0 0

2
1  (1)  1 
 5  
    (1)    
log(1  2 ) 
16 
4 2
8
 8  2 2
 



i 2 Li2 i (1  2  Li2 i ( 2  1

=
2 k ( k !) 2

2
k  0 ( 2 k )!( 2 k  1)
 /2
=
2
0
.15058433946987839463... 
1
 sin 1  cos1 
2

 
=

 

( 1) k k


k  1 ( 2 k  1)!

 1  dx
2 5
x
 sin x
1
.15068795010189670188... 
x sin x
dx
2  cos2 x
3 ( 3)  4
 2 log 2 2
log 4 2


 2 log 3 2 

4
15
4
4
 1  21
  6 Li4   
 (3) log 2 
 2
4
1

 
.1507282898071237098...
=
log 3 (1  x )
0 x 2 ( x  1) dx

8 log 2  4 log 3  1 
(1) k

k
k  1 3 k ( k  1)
1

2

csch 

4 2 6
3
(1) k 1

2
k  1 3k  2
.150770165868941637996... 

.15081749528969631588... 


  (k )  1 (k  1)  1
k 2
Berndt 3.3.2
(1) k
.15084550274572283253...   3
k 0 k  3
110
1

1
.1508916323731138546... 
   ,2,0   Li 2   
729
 10

 10 
2

113
k

.15097924480756398406... 


 
2
3
36
k 1 ( k  2)( k  4)

1 .15098236809467638636... 



 
=
 3
3 log 3 2 log 2


10
10
5
1
log(1  x 6 )
dx 

x6
0



1


2
k 1 6 k  5k
1 
1


2
4 k 1 9k  9k  2


1
 ( k  1)(6k  1) 
k 0

1
dx
  2

2
12 3
k 1  2k 
0 ( x  3)
4 
k 
7
7
1
6 1
 6 
.15116440861650701737...   (3)  Li3    Li3    log  log log 7  2 Li2    
6
6
7
7 2
 7 

H
 
=  k k 2
k 1 7 k

 ( 4 k  1)
.1511911868771830003...    4 k 1
1
k 1 2
 .15114994701951815422... 
81 .15123429216781268058... 

 8 

.15128172040710543908... 
 (1)
k 1
 0 (k )
3k
k 1
1
log10
2

1
.15129773663885396750...  

3
k 1 Fk k
1 .15129254649702284201... 
.151632664928158355901... 
1 .15163948007840140449... 
1
4 e
k 1



8
k2
k!2 k
coth
2


k
 (k  1)
k 2
2
csch 2

16
2
k 1  ( 2k )  1
=  ( 1)

4k
k 1

 
 (1)k 1
k 1

k5 / 2
k 1
.151696880108669610301... 


 ( 3) (5)
 (4)

2 .15168401595131957998... 

k2

k
k 1 10

5/2

4

25
4k 2

2
2
k  2 ( 4k  1)

.15169744087717637782... 
.151822325947027200439... 
.15189472903279400784... 

8
 /2
( 2 log 2  1)  
 
=
1 .151912873455946838843... 
2
x dx 
0
2
e 1
4  log 2
 

9 6
3
log


 log(sin x) sin
(1) k 1


2
k 1 2k  3k

1  dx

2  4
x
1

1

6
5
4
3
2
k 0 k  k  k  k  k  k  1

 log1  x

5
6
H
log   ( 1) k 1 kk
6
5 k 1
5

log  ( k )
.152020846736913608124...  
k2
k 2
.1519346306616288552...

.15204470482002019445... 
1+ 5
2 5 log
2 
2
.152185252101046135100... 
1
3
 cos 

5
2

6
 1  (2k  1)
k 1

2




4 log 2 2 log 2 2
Hk


 
18
3
3
k 1 ( k  1)( 2 k  1)

 2k 
16
1
    k
=
105
k  0  k  4 ( k  3)( k  4)
1 .15220558708386827494... 
.1523809523809523809
2

( 1) k 1

k 1  2 k 
  (2 k  1) k
k

.152607305856597283596... 
.15273597526733744319... 
17 .152789708268945095760... 

8
8
log 
7
7
1 e2


2 4

Hk
8
k 1
k

2k

k  0 k!( k  4)( k  7)
(  1) 2 

1 3 2
1
(2k )!!


arcsin
k
2
4
3 k 1 (2k  1)!!3

k 1
.15289600209100167943...   k 1 
k 2 k
1 .15281481352532739811... 
.15289756126777578495... 
e(e  1) 5
 
(e  1)3 e
k2


k
k 1 e


x2
1 e x dx
GR 4.384.9
.15300902999217927813... 
3   (2)   (3) 

1


4
3
k 2 k  k

1
 k (k  1)
k 1
3



 
=
  (k  3)  1
k 1
1 .15300902999217927813... 
4   (2)   (3) 

  (k )   (k  2)  2
k 2
1
1  1 


 1 
 
cot
  
   
 
2
2 3
3 2  3
3  



k 1
 (2k )   (2k  1)
 
=  3
 
3k
k 1 3k  k
k 1
log  (3)
.15309938731499271977... 
 (3)
.153030552913947039536... 

1 .15318817864647891814... 
1 . 15326598908047301786... 
 
1
si    1 
2  2
4
16

2
2
arcsin x
 arccos x dx
0



 

2
1
x 2 log x
dx 
2 0 x 2  1
Borwein-Devlin, p. 54



 

2 .153348094937162348268... 
( x  y) 2
x y
0 y x y sinh( x  y) log x  y dx dy
 coth  1  i  (1  i )   (1  i )   2 Im (1  i ) 


=
Borwein-Devlin, p. 53
 1
1 
  k  i  k  i 
k 1

2

k 1 k  1
3 .15334809493716234827... 
 coth   1  
4 .15334809493716234827... 
 coth   1  i  ( i )   (i )  2 Im (i )
.1534264097200273453...

=

1  log 2
( 1) k 1


2
k 1 ( 2 k  1) 2 k ( 2 k  1)


 2 k  1 
  1
 k log

 2 k  1 
k 1 

=
1
 k (8k  4)
k 1

=
2
2
k 2
1
k
(k 2  k )
J947
GR 0.238.2, J237
J128
( 2 k  3)! k
( 2 k )!
k 2

( 1) k
= 
k 0 2 k  4

( 1) k 1
= 
3
k  1 8k  2 k



1 
 ( 2k )

   1  4k 2 log1  2  
=  k
 4k  
k 1 4 ( 2k  1)
k 1 

=




=

1
dx

5
x  x3

=

0
=
 /4
 tan
1  sin 2 x
log x
1 x 2 dx 
3
log sin x dx

x
Prud. 2.6.34.44
dx
 log x  1  x
2

=
x dx
0
sin 3 x
2
K Ex. 109b
 1  dx
1 log1  x  ( x  1)2 
1

0
log(1  x)
dx
(1  x) 2
GR 4.291.14
1
=
 (1  x) arctan x dx
0
=
1
 1
1
1  dx
  

 
2
2 log x 2  log x
0 1 x
GR 4.283.5

=

 
=
 1 1   x e  x / 2  dx
     e 

 2 x
x  x
0

dx
0 e2 x (e2 x  1)
GR 3.438.1
k4
1 .15344961551374677440...  2 J1 ( 2)  3 J1 ( 2)  J 2 ( 2)  J 0 ( 2)   ( 1)
(k!) 2
k 0

(1) k 1  2k 
 
= 
k 1 ( 2k  1)!  k 
2
.153486153486153486
=
13

  1 1 / 3 
 7  (7) 2 / 3 
1 
1       2 1  1   


.1535170865068480981... 



i

i

1
3
1
3
1/ 3


  7 
7
6  71 / 3 
 7  






 (3k )
1
 
=  3
  (1) k 1 k
7
k 1 7 k  1
k 1

k






k2
.153540725195371548500...   2
1 
k 1 k  1  k
.15354517795933754758... 
 (1) (7)   (2)  H ( 2 ) 6   (2) 
1 .15356499489510775346... 
e1 / 7
1 .15383506784998943054... 
 1/ 8
5369

3600
181  log 2
 
 4 (3)  6 (2)  360 (1) 
90 5
2

 (k )  1
 
= 

k 2 k  4

k (2k ) 
 
  2 
12k 2
.15409235403694969975... 



   3 sin
 csc

k
2
2
48 
3
2 3 k 1 12
k 1 (12k  1)


2
9
k 1
.1541138063191885708...  2 (3)    ( 1) ( k  1)(k  2)( ( k )  1)  
3
4 k 3
k  2 ( k  1)
.15386843320506566025... 

=
  ( 2) 3

=
log 2 x
1 x 4  x3
1
=
x 2 log x
0 1  x dx

=
x2
0 e2 x e x  1dx

H ( 3) k

k
k 1 4 ( k  1)

.154142969550249868...

1
1
log 7  log 6  Li1     k 
7
k 1 7 k

1
1
 2  2 k 1
 
= 2 arctanh
(2k  1)
13
k  0 13
2

1

.15421256876702122842... 


64 k 1 (( 2 k  1)2  4 ) 2
.15415067982725830429... 

K148
J383

15 .15426224147926418976... 
ek
e 
k  0 k!
e

=
e
k 0
1 / k!
Not known to be transcendental
k 2
 ( k )  1 
k
k 2

2  1  
.15443132980306572121... 

 (k ) log k
k 1
k
2   
1 .15443132980306572121... 
3
2 .15443469003188372176... 


2k  1 
1
k 2
2
10

 ( 1)
.154475989979288816350... 

  2 log1 k   k ( k  1) 
k 1
log  (2 k  1)
k 1
2
40

 H ( 2)3 / 2
9
3

( 1) k
.15457893676444811...
  3
k  2 k log k

1
1 .1545804352581151025...  
k 1 S 2 ( 2k , k )

1 .1545763107479915715...

7
( 1) k k 3  2 k 
  k
 
32 2
k  1 4 ( 2 k  1)  k 

1
.15468248282360638137...  

3
k 1 (3k  1)  1
.15467960838455727096... 




 =

 .1547005383792515290...

 

9
18 3






 4 3  5  i 3   2 3  3i   5  i 3  
 6 
 6 






k 1

2
( 1) k  2 k 
1   k
 
3
k  0 2 ( k  1)  k 
log 3
1

6
18 3  3i







k 0




=
(2k )!

2 2k
4
 ( k!)
( 2 k  1)!!
 (2 k )! 4
k 1
k

2

1  2k 
 csc   k   
3
3
k  0 16  k 

(2k  1)!!
= 1  

2k
k 1 ( 2k )!!2

13
1
 
=

84
k  0 (3k  1)(3k  10)
1 .15470053837925152902... 

 
.15476190476190476190
1
2 .15484548537713570608... 
3e  6 
e
0
x 1/ 3
dx
AS 4.3.46, CFG B4
J166
K134
k4
7 .154845485377135706081...  3e  1  
k 1 ( k  1)!

k (k  1)

8 .154845485377135706081...  3e  
k!
k 1


.15491933384829667541... 
.15494982830181068512... 

.15555973599994086446... 
.15562624502848612111... 
 
=
1 .15572734979092171791... 
 (1)
k 1
k 1
 2k  k
  k
 k6
 Re(i ) 
   1   2   H 
2
 (2)   (3)
 (2)   (3)
.155761215939713984946... 

Hk
 (4k  1)(4k  1)
k 1

k 
2
k 1
1
2
k
216  18 3   2  54 log 3  72 log 2   (3) 


 (k  3)
1

(1) k 1


4
3
6k
k 1 6k  k
k 1



cos x
dx 
 
e   (1  x 2 ) 2

 (1)
k
 (k )  1
(k!) 2

2 cos t
 1 t
2
dt
0

1

  k  1  J  2
0
1  
 
k  

1
2
 1
.15580498523890464859...  4  4 2 arctan
 2 log  Li2    
 2
3
2
k

1

( 1)
 
=  k 2
k  1 2 k ( 2 k  1)

2k
k
.1559436947653744735...  cos 2   ( 1)

( 2 k )!
k 0

dx
.15609765541856722578...   4
x  x3  x
1
k 2


3  12

.154951797129101438393... 
log 2   G 
8
4
 2
.15519690003711989154...      
 3
.15535275300432320321... 

1 3

5 5

.1561951536544784133...


 (k )  1
k 1
H  (k )
 k k 
2
k 1
k 2

.15622977483540679645... 

2
k 2

1


1
 4k 1  2k  2 log1  k   2k
k 1

GR 1.411.3
2
 1 
log1    
 k 
7 

.156344287479823877804...  
 coth 
10 4
2
 ( 1)
k 1
 (2 k )  1
4k
k 1

e2
1  coth 1
1




2
2k
e 1
2
k 0 e
1 .15651764274966565182... 
.15652368328780337093... 
1

1
coth 1  1
1




2
2k
e 1
2
k 1 e
.15651764274966565182... 
6 cos1  10 sin 1 
23

2

k 1
.156643842100387720666... 

 

2

2
k 0
1

Bk 2 k 1

k!
k 0

1
k 
2
2
J951
1
(1) k 1


k 1 ( 2k )!( k  2)

1
e ee  ee

e  e cos 2 cos(sin 2)   
2
2
4
1 .15654977606425149905... 
1
k 
2i


k
 
 i  
 
= 
  Re  (k )  1  
2
k  2
 2  
k  2 4k  1


cos 3 x
cos 3 x

.15641068822825414085... 
dx


dx
3
2 2
2


e
0 (1  x )
 x  1



log 3 2   log 2 2  2 1 log 2 
2 i

sin 2 k

k!
k 1

( 1) k 1
log 2 k 

k
k 1

  1 = Stieltjes const.


12 .1567207587610607447...

 3  6 
x
3
sin x dx
0

1 .15688147304141093867... 
.156899682117108925297... 
.15707963267948966192... 
.157187089473767855916... 
2  log 
1 

   log x log 1  2 2 2  dx

 e x 
e
0
 3
3


4
6



20
3
3
e 1
e
5x
0


1  dx
3
 x3
 log1  x
1
dx
 e5 x

log( x 2  3)
3 .157465672184835229312...   log(1  3 )  
dx 
x2  1
0
1 .15753627711664634890... 
1 .15757868669705850021... 


26 (3)
1
1

 G3  1  


3
3
27
k 1 (3k  1)
k 1 (3k  1)
  3
  
4
 2

x 2 dx
e
0
x2
1
.157660149167832330391... 
=
.1577286052509934237...

.15779670004249836201... 
.15783266728161021181... 
9 .157839086795201393147... 
.157894736842105263
=
.15803013970713941960... 
.1580762...

1  2 2
 1 

 (1)    
9 6
 3 

x log x
1 x  x
2
dx 
GR 4.233.3
1


1
 
 (1)1 / 3 Li2 ((1)1 / 3 )  Li2 (1) 2 / 3
1/ 3 
 1  (1) 
2k
1 
3
3
k 3
cos 1   ( 1)

4 k 0
( 2 k )!

k 

1  k  

2 
k 2 
2

 1
k 1 k


(
1
)

 2e1 / 
k! k
k 1
8 log  
3
19

1
1
 1  dx

  cosh 4  9 
x  x
4 4e
1


GR 1.412.4
1
2
ω a non  trivial (  1)

integer power
.158151287891164991627... 
3 (3)
  ( 2) 
2
.15822957412289865433... 
 3 (2k  1)

k 1
x log 2 x
0 (1  x)2 dx 

log 2 x
1 x(1  x)2
Hk
k

.158299797338769411097... 
1
k!
 S ( 2k , k )
k 0
1

1 
(1) k 1

 csch   2
.15871527483457111423... 

2 4
2
k 1 4 k  1
i
i 


k 1  ( 2 k )  1
.158747447059348972898...   log  2    log  2     ( 1)
2
2  k 1
4k k



1  3
(1) k 1
.158806563699494307872... 


csch  3   2
6
6
k 1 k  3
2


1
log x dx
  4

2
16
x 1
k 1 ( 4k  1)
1

k
.15888308335967185650...  6 log 2  4   k
k 1 2 ( k  1)( k  2)

( 1) k
4 .15888308335967185650...  6 log 2  
1
2
k  0 ( k  3 )( k  3 )
.15886747797947540615... 

G

2
J125
k 2Hk
12 .15888308335967185650...  8  6 log 2  

k
k 1 2

k2
40 .15890107530112852685...  
k 1 Fk

B2 k  2 k 

1 .1589416532550922853...

 (2 k )!  k 
k 0

3 .15898367510013644053... 
7 .15899653680438509382... 
t4 (k )


k
k 1 2
I 0 (2 3 ) 


 2
i 1 j 1


 
=

1
kj
3
k
 (k!)
k 0
16
.15909335280743421091... 


14 245
.159105544771044128966... 
 0 (k )

2

1


   
1
1


ijk
iijkl
 1 i 1 j 1 k 1 l 1 2
j 1 k 1 2
 
11
 o F1 ;1; 3 
1
x
0
6
arcsin x dx 

GR 4.523.1
 

2 log 5  1  5 log 5  2 5 
5 5


Fk Fk 1
Fk Lk



k
k
k 1 4 k ( k  1)
k 1 4 ( 2k  1)

5
1
( 1) k 1
arctan  1   2 k 1
2
2
( 2 k  1)( 2 k  1)
k 1 2

13  16 log 2
1
.159137092586739587444... 
 
12
k 1 k ( 2k  1)( k  2)
.15911902250201529054... 

.1591484528128319195...

 ( 1)
k 1
k 1
9
1 
H2 k 1 
 4 log  2 2 arctan

k 
8
9
8
2 2
1
2
 (3)
1 .15924845988271663064... 
 (5)
.159154943091895335769... 
.159343689859175861354... 
.159436126875267470801... 

 (2k )   (2k  2) 2
k 1
2k

2
32

5 2 log 2 log 2 2



12
3
3
4 2
1
2
  (1)     (1)  
13 .159472534785811491779... 
3
 3
3
2
4



k 1 cos k
.15956281008284001011... 
  1   (1)
24 2
k2
k 1
( 1) k 1 Hk

k  1 k ( k  3)

.15972222222222222222
=
23

144

1
 k (k  1)(k  4)
k 1

e
1
1

log coth   k
2
4
k  0 e ( 2k  1)


1
 (k )
2
k log k
.159868903742430971757...   log 2  log 2   ( 1)
  k
2
k
k 2
k 1 2 k
(1)  5 
2 .15990451200777555816...     
 6
1 .15973454203768787427... 
e arccoth e 
.160000000000000000000 =
(1) k 1 k

4k
k 1

4

25

1 .160014413116984775294... 
.1600718432431485233...

2 .1601271206667850515...

.16019301354439554287... 
.16043435647123872247... 
3 1
sin 3 x
 
dx
4  3  0 x 3

1
 (k )
(   log 4  log 5)   ( 1)k k 
5
4
k 1

log k log(k  1)

k2
k 1


(1) k 1
H
 1
 Li2      k 2   k k
 6
k 1 6 k
k 1 7 k

16 7 log 2
1

  k
9
3
k 1 2 k ( k  3)


.16044978576935580042... 
2 3e
.1605922055573114571...
.1606027941427883920...




3
cos x
dx
2
3
x
0

24 log 2  7
1
 2

60
k 1 2k  11k  12
2
1
x
0
4
 1
log1  dx
x


( 1)
log 2 x
2 x


x
e
dx

0
1 x 2 dx 
k  0 k !( k  3)

5

e
k
1
( 1)k

k  0 ( k  1)!  2 k !

 sin 1 1
cos 2k
  
.160889766020364044721... 

4
2
k 1 ( 2k  1)( 2k  1)
 
.1610391299195894597...

=

 2
2
 log 2  4 
GR 1.444.7
2
2 2
log

2
2 2

 
=
(1) k 1

2
k 1 4k  k

 
=
1  dx

4
0 log(1  x ) dx  1 log1  x 4  x 2


1
4 .1610391299195894597...

 2
2
.1613630697302135426...

k 1
(1)

k
k 1
1
coth
2

.16126033258846573748... 
2
2 2

 log 2 
log
2
2 2
1

1
4  dx

 log1  x
0

 1 1
log 2
log k
 Lik      
 k
2
k 1 2
  2 2


1
1
B2 k 2 k
1  2 2
 
1
2
k 1 k   2
k 1 ( 2 k )!
 ( 2k )  1

.16137647245029392651... 
  2 

 2 
  J0
  2 
0
k
 k


1
 (2k )!(2k )!  2   I 
k 1
.161439361571195633610... 
=
=
1 .161439361571195633610... 
 e2  1 
 

log sinh 1 log
2
e


2 k 1

2
B2 k

k  1 ( 2 k )! k
22 k 1 B2 k

k 1 ( 2k )!k


 
i 
i 
(1) k 1 (2k )
 log 1   1     
 2k k
k 1
      

 e2  1 
(1) k 2k Bk
  
1  log sinh 1  log
k !k
k 1
 2 
3
Berndt 5.8.5
1
arctan 2 x dx
 
.16149102437656156341... 

192
1  x2
0
.161745915929683120084... 


=





2 6
 log 17  12 2  3 arctan h



1

3


5
12 2 

1
x
6
1
1
dx
 x 6

2 .16196647795827451054... 
1 .162037037037037037037 =
kk

k 1 2k 
 
k 
251
 H ( 3) 3
216

1 .16204051536831577300... 
 (k )  (2k  1)  1
k 2

(1) k 1
 1 
.16208817230500686559...  cos
 1  
k
k 1 ( 2k )!3
 3

3 3
1
8 .16209713905398032767... 
 
1
5
2
k  0 ( k  6 )( k  6 )

6
1
.162162162162162162162 =

  (1) k e  ( 2 k 1) log 6
37
2 cosh log 6
k 0

=
2 .1622134783048980717...

3 .162277660168379331999... 
sin 6 x
dx
ex
0

2
2

1

2
k 1 k ( 2k  1)
 4 log 2  2
10
k (k  2)
2k
k 1


J943
1 .16231908520772565705... 
.162336230032411940316... 
.162468300944839014604... 
1
 
3
 , 0,1  

 2
 k 1 (k  12 )!

71
k3
 
13e 
2
k 1 ( k  3)!
 4  93 (5)
6
 (1)k 1

 
=

k
 (k 

k!
1
2
)5
1
 2k 1  2k  2k e

2 1 / k
2
k 1
 2k 2 
8   27(2k  1()1) 2(2k  1) 

2 1
k
3
k 1
1


 (k )
k 3
.16266128557107162607... 
2e

k 1

.162651779074113703372... 
2
e(erf 1) 
x arcsin x
 (1  x )
2 2
dx
GR 4.512.7
0
2
.16268207245178092744... 
.16277007036916213982... 
.162824214565173861783... 
.162865005917789330363... 
1 .16292745855318123597... 
1
dx
  4

2 log 2  log 3 
8
x  x3
1

1
4
( 1) k 1
2  3 arctan  2 log   2 k 1
2
5
( 2 k  1) k ( 2 k  1)
k 1 2
1

 3
1
11 2 log 2 2
x log(1  x)

 
dx
96
8
1  x2
0
 3  3 (3)
256


 
=
 ( 1)
k
k 0
.16301233304332304576... 
.163048811670032322598... 

4e
1  (2)  7 
 5
 3
 1 
     ( 2 )     ( 2 )     ( 2 )    
 8
 8 
 8
 8
1024 

1
1
1
1


3 
3 
3 
3
 (4 k  1)
( 4 k  2)
(4 k  3)
( 4 k  4) 


 xe
 x2
sin x cos x dx
0

(k )
k 1
k


log (2 k  1)  

.1630566660772681888...
.163224793...
k2
7  3e    Ei(1)  
k 1 ( k  2 )!( k  1)

( k )
 
2
k  2 ( k  1)

.16329442747435331717... 

 
=
4 log 2
2
8 
 ( 2  1) 

log(1  2 ) 
3
3
9 6

1

k  1 k (8k  3)
Prud. 5.1.7.22
.163265306122448979592... 
.16345258536674495499... 
.163763357369443580273... 
.16389714773777593436... 
.163900632837673937287... 
.16395341373865284877... 
2 .16395341373865284877... 
4 .16395341373865284877... 
.16402319032763527735... 

 
=
2 .16408863181124968788... 


 .1642135623730950488...

1 .16422971372530337364... 
.16425953179193084498... 
8

49

k
8
k 1
k
2
40


6
27
 /2
x
2
cos3 x dx
0
17
  (2)  3 (3) 
8

k
 (k  3)
k 1
3

 3 
(1) k 1 k
1 2 e


   
cos
3
(3k )!
9e
9
k 1
 2

  3   5 
3
 ( 2k )  1
log
 log       

k
8
  2   2   k 1 4 k

3 e
1

  2 2
1
e 1
k 1  k  4

1 e 1
B
coth 
  2k 
2 e  1 k  0 (2k )!

3e  1
1

  2 2
1
e 1
k 0  k  4
1  (1)  2 
3 2
1
 1
(1)  4  


  (1)   



 
 

 3
 3 
 3
12 
4
9
6

9k


2
2
k 1 (9k  1)

 (k )


2
k 2 k  3

( 2 k  1)!!
5
2 
 

k
4
k  2 ( 2 k )! 2
 4
 
5
 2  log 2 2
 1  i
 1  i
log 2 
 
 Li2 
  Li2 

 2 
 2 
24 4
2
log(1  x 2 )
0 x(1  x) 2 dx 
1
=


1 .16429638328046168107... 
1
 (k!!)
k 1

.164351151735278946663... 
.164401953893165429653... 

1
3

3 3
log 2
3

2


log 2

3

(1) k


k  0 3k  4

(1) H k
k
 2 (k  1)
k 1
k

x
1
5
dx

 x2
6 .16441400296897645025... 
.16447904046849545588... 
38 
1
   Ei(1)  1 
e
1
=

0
1 .16448105293002501181... 
.16449340668482264365... 
2 .16464646742227638303... 
.164707267714841884974... 
.16482268215827724019... 

(1) k 1 k


k  0 ( k  1)!( k  1)

x log x
dx
ex

1
1
 log 2 2  2 Li2     k 2 
6
 2  k 1 2 k

2
1  dx

  log 1  5 

x  x
60
1
2
4
45

H ( 2) k

k
k 0 2
log(1  x) log 2 x

x
0
1
 2 (4)   
7 (3)
 2 log 2  4
(1  log 2) 


8
8
128
 (3)
1  log k



 (3)
 (3) k  2 k 3

(1) k


2
k  0 k!( k  2)


p prime

kH k
 (2k  1)
k 1
log p

p3  1
(k )
3
k 1 k


 
=
.164895120905594654651... 
1 .16494809158137192362... 

 (3) 1786050 3) (6)


 (6)
 12

1

4

.165122266041052985705... 
 ( 1)
k 1

 (k ) log k
k 1
k2

(k )
6k  1
.165129148016224359068...   (3)   (5) 
k 1

9 .1651513899116800132...

84  2 21 

k 1


2
k  2 k ( k  k  1)

2 .16522134231822621051... 


 F  (k )  1 
k 2
k


 5  1 
 5
3
5


    

5
5
tan

2
2
2
 2  10

1 2
3
(1) k 1
e cos
.16528053142278903778...  1 

 

3e 3
2
k 1 (3k )!

Hk
2 .1653822153269363594...  e  Ei ( 1)   
k 1 k !
.16542114370045092921...    (1)
 
=
3
k 1
 1   (3)   3

k  2 k log k

.16546878552437427...
1 .16548809348122843455... 




1
log n k

 k3
n 1 ( n  1)! k  2
1
I 0 (1)  L0 (1)  e    e x arcsin x dx
2
0
.1655219496203034616...

1 .16557116154792148338... 
2log 2 3  4 log 2 log 3  4 log 2 2 


4


 
=
x
2
1


4
k 1
k
Hk
( k  1)
   6(k1) 1  6(k1) 3  


3
log 2  3 
6

k
k
k 0
dx

 x 2  1

.165672052444307559...

1
2  arctan 2  log 5    e x sin x cos x log x dx 
10
0

1


 2k  2 
k 1
1
 2 log 2 log3 2 1  1 
.16585831801671439398...  1 

 Li3     1  Li3 (2) 
12
12
2  2
2
6 csc  e sin  2 
3 .165673248894309178142... 

 1  k
2


 
log 2 x dx
=  3

x  2 x2
1
.165873416657810296642... 

 
=
.16590906347585196071... 
16 .16596749219211504167... 
2
1
cosh1sin 1  cos1sinh 1  cos1  sin 1  e (sin1  cos1) 
4
8e
k 1 k

(1) 4 k

(4k )!
k 1
9 log 3  2  3
63




50
30
50
250
1
 1
  ( 2)   
 4
8

 3k
k 1
3
1
 5k 2
2
log x dx

3/ 2
 x  1/ 2
x
1


k2
.16599938905440206094...   ( 2)  4 (3)  4 (4)  1  
4
k 1 ( k  2)
.166243616123275120553... 
1 .166243616123275120553... 
.166269974868850951116... 
2
 (2k  1)!! 
1


 1   

k 1  ( 2 k )!!  ( 2k  1)

4G
4G

J385
2
 2k 
1
   
k
k  0  k  16 ( 2k  1)

1  cos1cosh 1 
(1) k 1 4k


(4k )!
k 1

GR 1.413.2
log 3 2
.16651232599446473986... 
2
.16652097674818765092... 
( 2) 

.16662631182952538859... 


2

4

log 2  log 2


2
8

.166666666666666666666 =
=
=
k2 1

4
3
k 1 4 k  2 k

2k
k 2
2
.16658896190385933716... 
 3 log 2 
2
( k )  ( k  2)

=
( 3)
 /4

0
x sin x
dx  GR 3.811.5
(sin x  cos x) cos 2 x


1
k
1
4

 
  2 2   1  2 
6
k 
k 3 
k  3 ( k  1)!
k 1 k 

 ( 2k )  1

4k
k 1


1
k



3
k  2 k  k  1/ k
k 1 ( k  2)( k  3)( k  4)
(1) k

k  0 ( 2k  1)( 2k  5)

=

1
(1) k 1  2k 

 


k
k  2 k!( k  2)
k 1 2 ( k  2)  k 

4

1  2 

k 
k 3 


k ( k  4)
k ( k  6)



k 1 ( k  2 )( k  2 )
k 1 ( k  5)( k  1)

arccosh x
1 (1  x) 3 dx

=
=
=
=
1 .166666666666666666666 =
 2 (4)
7


 (8)
6
=
 1  p 4 
   4 
p prime  1  p

2 ( k )

4
k 1 k
J1061

HW Thm. 301
Berndt Ch. 5

sinh 1  sin 1
e
1
sin 1
1



 
2
4 4e
2
k  0 ( 4 k  3)!


  2 
 (k )  1
1
.16699172896087386807...  
   I 0 
  1  
k !k!
k
 k
k 2
k 2 
.16686510441795247511... 
1 .16706362568697266872... 



1
4k
cos 2  cosh 2  cos (1)1 / 4  cosh (1)1 / 4   
2
k  0 ( 4k )!
2 k (k  1)
22 .167168296791950681691...  2e  

k!
k 0

2
arccos
2 .16736062588226195190... 
cosh  2  cos  2

2 2




1 .16723171987003124525... 
8
GR 1.212


 
sin (1)1 / 4  sin (1)3 / 4 


1
4 


k 1
3 1
 11 
5
    
.167407484127008886452...    log 2  3 log
3 1
 12 
6
1

k

(1)
 
= 
2
k 1 6k  k
 
=
.16745771316555894862... 
5 .16771278004997002925... 

2


k 1


 1  k

1
 1  2(k  1)! 


3
, volume of the unit sphere in R6
6

  log 
( k )
.16782559481552120796... 
 log 2   ( 1) k k 
2
2 k
k 2

2 k  1
 1
= 
 log


2k 
k 1 2 k

9
3
1
1
.16791444558408471119... 
 cot   2 2
1
2
2
3
k 1 k   9

.16805750730968383857... 

k 1
 (k )
k
3

1
p prime 3  1

p
1
2
3
e 
 
cos
3 
2 
e

((k  1)!)3
.16807806319774282939...  

(3k )!
k 1
1 .16805831337591852552... 

.16809262741929253132... 
 ( 3)  1

 ( 3)

1 .168156588029466646...

log 2 k

k  2 2 k ( k  1)
 

1 .16823412933514314651... 
k 1
4
Dingle 3.37

(3k )  1

1
 (3k )!
k 0
J803
.16823611831060646525... 
arctanh

1

6
6
k 0
2 k 1
1
(2k  1)
AS 4.5.64, J941

 5  i 3 



 4 
1
 log 2
  

2

6
3
6
i

3 3i  3
3
3
3








1
 ( 3k )
  k

3
8
k 1 ( 2 k )  1
k 1


2k
  (1) k 1 2k  (3k  1)  1

3
k 2 k  2
k 1

1
1
 
Li4     k 4
6
k 1 6 k
3 log 3

1
1
 2 log 2 
      
2
2 3
 2
 3

1
2
2
x  x 1
1 x
1 log 1  x  x 2 dx  0 log 1  x 2 dx
 5  i 3

 4 

.16838922476583426925... 

 
1 .168437795256228934...
=

.16846317173057183421... 
1 .1685237539993828436...

 

=
.168547888329363395939... 
.168609905834157724449... 
.168655097090202230721... 


2
 
=



1
 2(2 k  1) 2
4 16
k 1

1
(1) k 1 2k
2  log 216  7   
 (k  1)  1 
4
k 1 ( k  1)( k  2)



4 2 log
 8(2 k  1)

2 1 



 8(2k  1)
k 1
4
2
4
4
 2


2  1  8 log 2 
1
 2(2 k  1) 3
1 
1 
1 
4  2
 
    1 
   1 
2  2 2 2 
4 

1
= 
3
k 1 8 k  k

 (k )  1
.16905087983986064158...  
kk
k 2
 1   1 
  
1 .16936160473579879151...  

1 / 5   4 / 5 
.16870806500348277326... 
.16944667603360116353... 
(1) k

3
k 2 k  3
.16945266811965260313... 
14 
 
27 9

1
x
0
7 ( 3)

16
2
arcsin x arccos x dx

 (2 k  1)
k 1
8k


.16945271999473895451... 
.16948029848741747267... 
=

=
  (4k )  1
k 1


  (8k  4)  1

k 1

1


12
 k 4
k 2 k

  (16k  4)  1
k 1


k
k 2

2  cot
12  22  6 2   2 

0
  (16k  4)  1 
k 1
Berndt ch. 31
x 3 log 2 xdx
ex

 2k 
3
3
1
arcsin
    k
2
2
k  0  k  6 ( 2k  1)

1
k 1
.17005969112949392838...   2 log
k
k 2 k

1
1
.17032355274830491796...  Li3     k 3
6
k 1 6 k
1
2
x dx
 1
.170557349502438204366...  1 
 log(e  1)  Li2      x
12
 e
0 e 1
1 .17001912860314990390... 

2 .17063941571244840982... 

 (k )  1
k 2

23
 5 / 4 cot  23 / 4  coth  23 / 4 
14 2


8
 
=  4
  8k  (4k )  1
k 2 k  8
k 1



 1
 ( 2k )
1
.1706661896898962359... 
coth    2
  (1) k 1 k
6
3 2
9
k 1 9k  1
k 1


log 3
5 1 (1)  2 
6k  1
.1707701846682093606... 


     

2
2
6 3 12 9
 3
k 1 3k ( k  1)

k ( (k )  1)
 
= 
3k
k 1

1 .17063957497359386223... 


3
F
.17083144025442980463...   kk 
8
1
 k4


11 .1699273161019477314...
20

1
1
 2 2

1
2
k 1 k   4

1
1
 
=  k tan k 1 
2
k 1 2
3 .1699250014423123629...  log 2 9
.16951227828754808073... 

 3  
1 
2
e
cos
   
9 
2
3  
e


( k 4 )

k4
k 2


1
1




4
4
4
k 2 k  1
k 2 k  k


.1710822479183635679...


( k )
4k
k 1
.1711785303644720138...

2 .17132436809105815141... 
1 .1713480439548652889...

.17138335479471051464... 
1
  8 

Re ( i )  i (i )
cosh
1

3

1
 3 (2k )! 
k 0
k
 (1)  (k  1)  1
k 1

1 .17139260811917011128... 
 (3k  1)  1 

 (k )  1 

k 2
1
8
4
( k  1) ( k )  1 

d (k )k  4
Titchmarsh 1.2.1

8100
k 1

1 5
cosk

.171458091265465835370...   log  cos1   k 


2
4
k 1 2 k


 2k 
1
(2k  1)!!
.1715728752538099024...  3  2 2     k


k
k  0  k  8 ( k  1)
k 1 ( 2 k )!2 ( k  1)
1 .1714235822309350626...

  2 ( 4) 
( 1) k 1 (2 k  1)!!

= 
(2 k )! ( k  1)
k 1


 
=
1  (1) k
(k  2)

2 k  0 (2k )!

.17166142139812197607... 

 (2k )  1
k 1

.17172241473708489791... 
.17173815835609831453... 
( k  1)
2



  k
k 2
2
 1 
Li2  2   1 
k  
 (3)
7
6  3 cos1 5 sin 1 
e

 2 k  ( 1) k 1
  k

k  1  k  4 ( k  1)

1 
(1) k


2 k  0 (2k )!(k  2)
log3 x cos log x
 
= 
dx 
x
1
1
.17186598552400983788...    1   ( 2  i )   (2  i ) 
2
1
1
=    (i )   (i )  1     (1  i )   (1  i )  1
2
2


1
  ( 1) k 1  (2 k  1)  1
=  3
k

k
k 2
k 1

=
  (4k  1)   (4k  1)
k 1

=
1  cos x
dx 
x
x
 1
 e e
0

.17190806304093687756... 
x
4
1
1 .1719536193447294453...
dx
 x2  x
 




i
x dx
 GR 3.418.1
Li2 (1)1 / 3  Li2  (1) 2 / 3   x
x
e

e

1
3
0


=
1  (1)  1 2 2 
log x
   
  
dx
 3
3
3 
1  x  x2
0
1

 
5 .172029132381426797...

1 .1720419066693079021...

 2  9 (3)
4
GR 4.23.2
1
log 3 x
 
dx
3
0 ( x  1)

8
sinh 
5
2


1
 1  4(k  1)
k 1

3




log k
2
2
k 2

1 1
.17210289868366378217... 
 tanh 
2 2
4
2
3 .1722189581254505277...  e
1 .1720663568851251981...

k

 (1)
k 1  k / 2
e
J944
k 1

.17224705723145852332... 


 .17229635526084760319... 
k2

k 1 (3k )!
 ( 2 )   ( 3) 
( 4)


Hk
 k ( k  1)
3
4
k 1

k
2 4
.17240583062364046923... 
   k  (2k )  1 
16
9
k 1 4

1
1
1
 .17250070367941164573...  1 
cot
  2 2
1
2
2
k 1 k   2
1
  1 1
.172504053920943245495... 
 3 log 3     li  sin(3 log x) dx
10 
2
 x
0

k3


3
2
k  2 ( k  1)

1
2 .17258661797937013053...   3
k 1 Fk
.17254456941296968757... 
.17259944116823072530... 

 (1)
k 1
k  (3k )  1
k 1
3
3

2 3
 3 log  4 log  2  
2
2
2


Hk
 3 ( k  2) 
k 1
k
GR 6.213.1

 ( 2k )

2k
k
k 1 
(1) k 1 B2 k 4k

(2k )!2k
k 1

.17260374626909167851... 

 log sin 1  
AS 4.3.71, Berndt Ch. 5
2 .172632057285762871628... 
 ( ( 2 ))
 (k )

.17264325482630846776... 
.17273053919054494131... 
.1728274509745820502...

 (1)
k
3
k 2
k
5 5
 
e 3

 log 2
2



1
 1
  Li   k   k 
k 1
(1)

2

k 1
 k!(k  3)
k 1
( 1) k 1 Hk
G  
2k  1
k 0

Adamchik (24)
log(1  x 2 )
0 1  x 2 dx
1
=
.172855673055088999098... 
4e  

x

2
GR 4.295.5
cos 4 x
  2 / 16

.17289680030426476...

log k

k
k
k 2
2

1 .17303552576131315974... 
 (2k ) (2k )  1 
k 1
 
=

 ( 2k )
k 1 n  2
n
2k



1

k 2

1
 m k
m 1 k 1

  2  2k cot k  
2
2
1

 k2  3
3


  
 cot 
2
4 k  2  2(k  1) k
k

2
2 Ei(1)
(1) k 1 kH k
.17305861469432215834...   
 Ei(1) 
 
e
e
(k  1)!
k 1
=

.17311810051999912821... 
  ( x)  1 dx
2
2
2 .17325431251955413824... 
 (3 / 2)
1  n

 lim
  h( m)  n 
n


 (3)
n  m 1

h(m) = lowest exponsent in prime factorization of m.
.173280780307794293319... 
1 


1
  2  ei     2  e  i  
2

 
=
  ( k  1)  1 cos k 
k 1

.17328679513998632735... 
log 2 
1


4
k 1 ( 4k  3)( 4k  2)( 4 k  1)
J253
(1) k

k  0 4k  4

=

=

1
 (4s  1)
r 1 s 1


 
=
 8(2k  1)
k 1

=
x
3

2r
J1124
1
 2 ( 2 k  1)
[Ramanujan] Berndt Ch. 2
dx
x
5
1
1
=
x log x
dx
2 2
0 (1  x )


1 .1733466294835716297...

1
 k!!k
k 1

.17338224470333561885... 

k !k

 



2

3
4 log 2  3 
1 1

2
 (k )  1
k 2
3 . 17343648530607134219... 
GR 4.234.2
dx dy

1 x2  y2
1 1

Borwein-Devlin, p. 53
(1) k

3
k  0 2k  3

.17352520779833572367... 
2 .173532954993580700771... 
log 2 (1  x )
dx
0
x
1 .17356302722472693495... 
  eEi(1) 
1

 (k  1)
k 1
k!



BernoulliB (2k )
EulerE (2k )
k 1
.173749320202926737911... 

.17378563457299202316... 
 (6k )
.17402964843834081341... 
1
 ( 4)
4
  (2) (3)  2 (5) 
 3 (3
k 1

kH k
 (k  1)
k 1

.17412827332774280224... 
k
k
1

 1)k
k
.17417956274165000788... 
1
Li2   
 6
.17426680583299550083... 
arcsinh1 

1


k 2
k 1 6 k
(1) k 1 H k


5k k
k 1



1
2
 log 1  2 
2
2
4
1
=
.17440510968851802738... 
x 2 dx
0 ( x 2  1)3 / 2
2
6
 3 3  9 log 3  6 log 4  36 

=
 (1)
k 1
 (k  2)
6k
k 1


e1 / k
k
k 1

1


.174644258731592437126...   1  2

k  1/ 2 
k 1 
2 .17454635174140...

 (1)k 1

.17472077024896732623... 
  (k )  1
4
k 1
.17476263929944353642... 

p prime
.17488106576263372626... 
 6k
k 1




1

p3

 (k )
k 1
k

3log 2
2 

4
2
log  (3k )

1
 k (8k  2) 
k 1
3
1

 k2
.1750000000000000000
1 .1752011936438014569...

 
=
7
40
e  e 1
1
1
 sinh 1 
 2 cosh sinh   isin i 
2
2
2

1
= 

AS 4.5.62, GR 1.411.2
k  0 ( 2 k  1)!

1 

GR 1.431.2
=  1  2 2 
 k 
k 1 

=
 1  dx
 cosh x  x
2
1
.175311881551605975687... 

3 log 2 

2
1

3

 (k  1)  1
k 1
4k

1 i
 2 F1 1,1  i, 2  i,  2  i 2 F1 1,1  i, 2  i,  2 
4

cos x
=  x
dx
0 e 2
.17542341873682316960... 

 

.175497994056966649393... 
log k
 (k  1)!
k 1

.175510690849480992305... 
e
k 1
.175639364649935926576... 
1
1
2k
1
e

2
3
.17575073121372975946...  8 

8e2
.175781250000000000000 =
45

256

k 1


k
k  2 k!2


1  log n k

 k
n 1 n! k  2 k!2
1
x sinh x
dx
ex
0


k2

k
k 1 9

  1
3
9
 2
1
 2
log 2 log 3  Li2   log  2 Li3     Li3    Li3     (3)  
2
4
 3
 3
 3

  2
( 2)

H k
k 1
=  ( 1)

k
2 (k  1)
k 1
.175886808246818344795... 

 

J 2 (2 3 )
1
(1) k e k
 0 F1 ; 3; e   
e
2
k  0 k!( k  2)!


 (k )  1
1
1 

   log1  
.17593361191311075521...   k
 2k 
k  2 2 ( k  1)
k  2 2k
.17592265828206878168... 
1 .1759460993017423069...

i  ( 1)  1  i 
 1  i
   ( 1) 
 
 
 2 
 2 
4

x sin x
 sinh x dx
0

.17606065374881092046... 
 6k
k 1
.176081477370773117343... 
12 .1761363792503046546... 
1 .17624173838258275887... 
1 .176270818259917...

1 .1762808182599175065... 
.176394350735484192865... 
1
2 3
4
1
1
3
sinh


 (1)
8
 ( )
 (3k )
6k
k 1
1

3

k
 (2k )!3
k 1


k 1
k

log 3 x dx
x 3 dx

0 x 2  1 0 sinh x
smallest known Salem number
larger real root of Lehmer’s polynomial
z 10  z 9  z 7  z 6  z 5  z 4  z 3  z  1


log 2
arctan x
1
dx


 
x6
20 20
10
1
Borwein-Devlin. p. 35
 (1) k 1 
1 

k

22 
k 1 

1 .176434685750034331624...

3 
log 8
 ( k  1)  1
.176522118686148379106... 

 
  ( 1) k 1
2 2
2
k 2
k 1

1 
 1

= 
 k log 1    1 

k 
k  2  2k
1  1/ 2
 log 2  1 
4 2
1  1/ 2

1
=  k

k  0 2 ( k  1)( k  2)( k  3)
3
.176528539860746230765... 

 
log
(1) k 1

k
k 1 3  1

.17659040885772532883... 
.1767766952966368811...

.176849761028064425527... 

4 2
( 1) k k  2 k 
 
4k  k 
k 0


e 1 / 
( 1) k
k 1 ( 2 k  1)!( 2 k  3)
e

3
log x sin log x
 1  dx
dx   sin  5 
= 
 x x
x
1
1
.17709857491700906705... 

1

5 cos 1  3 sin 1  
 

1 .177286727908419879284... 
k 2
2
(k )  1 k 1
J279


1
(1) k 1
  
.17729989403903630843... 
6 3 8
k  0 (3k  1)(3k  5)

1
27
1 .17730832347023263216... 
log 3  9 log 2  

2
4
k  0 (2 k  1)[(2 k  1)  1 / 9 ]
 H H

k
k 1
.17733145610396423590... 
24 log 2  9 (3)   2  12  
2
12
k  1 k ( k  1)



1
 1
4 I 2 (2)  0 F1  ; 3;   2
k
 2
k  0 k!( k  2)!2


(1) k
1
2
  2  
.177532966575886781764...  1 

2
12
k 2 k
k 1 k ( k  2)

cos 2k
1
=
Li2  e 2i  Li2  e  2i   (1) k
k2
2
k 1
1 .17743789377685370624... 
 




1

 
=
log x
0 xlog(1  x) log x dx  1 x 2 ( x  1) dx 


 
=
x
 e e
x
0
1

 
=

0
1 1

 
=
x
 1

e
0
x dx
 GR 3.411.10
 ex
2x
dx 
x log x
dx 
1 x
xy
  1  xy dx dy 
0 0

2 .1775860903036021305...
1

1

 
k 1
k
3 
2k  1
cos
J513
log(1  x)
dx
1  x2
log( x 2  1)
= 
dx 
x2  1
0
=

 
1
2 3 log(2  3 )   
8
  log 2
.17756041552698560584... 


log( x 2  4)
0 x 2  4 dx 

=
log(1  x 2 )
0 1  x 2 dx
GR 4.295.15


 
=
 1  x 2  dx
log
0  x  1  x 2
=
  x tan x dx

0
GR 4.298.9
 /2

=
0
x2
dx
sin 2 x
GR 3.837.1
1

 
=

 
=
arcsin x
1 x dx 

arctan x
dx 
2
x
0


1 .177662037037037037
=

1 .17766403002319739668... 
 x
  log sin  dx 
 2
0
2035
 H ( 3) 4 
1728
 /2
 log(4 tan x) dx 
0

7

1
1
 ci      log 2  
2 k 1
(2k  1)
 2
k  0 ( 2 k  1)!2
3

k
.17785231624655055289...  

k 1 (3k )!
.17778407880661290134... 


 .177860854725449691202... 
 2 ( 3)   ( 6)

2 ( 3)
1
n
3
, where n has an odd number of prime factors
[Ramanujan] Berndt Ch. 5
.17802570195060925877... 
log 2  2 



2
16 4

.17804799789057856097... 
e
k 1
3 .17805383034794561965... 
1 .17809724509617246442... 
=
=
=

0
x 2 tan x
dx 
cos 2 x
1
1
GR 3.839.3
Berndt 6.14.4
2k
log 4! 


3
 arctan 1  2 
8

( 1) k 1
6k  3
sin

2
2
k 1 ( 2 k  1)
2


( k  1)
4 k 2  8k  4



2
1
3
k 1 ( k  2 )( k  2 )
k 1 4 k  8 k  3

dx
0 ( x 2  1) 3
1
=
 /4

0
x 3/ 2
dx
1 x
J523
GR 3.226.2

=
.1781397802902698674...

sin 3 x
0 x dx
1 

.17814029797808795269... 
2
2
GR 3.827.4

2
24


log x sin x
dx
x2
0

Hk
k 2
k
6
k 1
( 2)

1
 ( k  1)
 
 
.1781631171987300031... 
2
4 2
k 1 k ( k  1)( k  2 )
2
 2

2k
e e
2 .1781835566085708640...  cosh 2 


2
k  0 ( 2 k )!

k2
10 .17822221602772843362...  
k  2 k !!
11 .17822221602772843362... 
.1782494456033578066...


e
 1 
erf 
2 
2
 2
2
1 
dx
  arctan 2   4
2 4
x  x2
1
3 e 3
2
 (3)

k2


k 1 k!!
(2k  1) 2
 
4
k  0 ( 2 k  2)


144
4

k !k !k !
.178403258246176718301...  
k  1 ( 3k )!

1
k2
.17843151565841244881... 
   log 2  
 (2 k  1)  1
16
k 1 k  1

25
1
 3e  
.178487847956197627252... 
3
k  0 k!( k  2)( k  5)

4
5
H
.1785148410513678046... 
log   ( 1)k 1 kk 
5
4 k 1
4


log 3
1
.1787967688915270398... 

 
4
4 3
k 1 3k ( 3k  1)( 3k  2 )

dx
=  4
x  x2  1
1
.17836449302910417474... 
.17881507892743738987... 
.17888543819998317571... 
24

4
tan 1
2 arctan
1 
2
2
5 5

1
sin 2 x
0 1  cos2 x dx
( 1) k 1 ( 2 k )! k


( k !) 2
k 1

GR 1.411.2
J250

2 .17894802393254433150... 
 2  (k )  1
2
k

k 2

.178953692032834648497... 
=

1 cot 1
1


  2 2
2
2
k 1 k   1

 (2k )  (1) k 1 22 k 1 B2 k


2k
(2k )!
k 1 
k 1

.17895690327368864585... 
1 .178979744472167270232... 
8


erfi1  erf 1   sinh x1  dx
x
4

2 si ( )


2 sin x
dx
 0 x
( 1) k 1 ( k  12 )!
  
( k  1)! 2 k
k 1

5  2 6 
.17922493693603655502... 
3
20
 16 log 
2
3

(1) k

k
k  0 2 ( k  4)

2  log 3 1
1  2

 log 1   
4
2e 4 
e

.17928754997141122017... 
 (1)
k 2
1 e
k
1  (  1) log(  1)  log   

145739620510

387420489


k 1
0
k


k 1
k
1
k ( k  1)
log 2  log 2
log 2 2
 




4
2
24
2
2
4
1 .17963464938133757842... 
1

 3

cot

6 4 3
2

 4k
k 1
J149

2
 e
=
k 10
 10
5 .179610631848751409867... 
.17974050277429023935... 
x
(1) k 1 ek
 

k  1 ( k  1)!
.17938430648086754802... 
.1795075337635633909...
dx

x
 2)
 e (e

e
2
1
 (k )  1

 (k )  1
.179374078734017181962... 
376 .179434614259650062029... 
3
1
.17905419982836822413... 
.1792683857302641886...
GKP eq. 6.88
1
2
3
1
3
3
3
3
log (2  3 ) 
arctanh

 
4
2
4
2
3

k

k
k  0 3 ( 2k  1)

x log 2 xdx
0 e2 x 
1
5
3      1    2 log 2 
 3
6

7
7
H
.179842459798468021675... 
log   kk 
6
6
k 1 7
9
.180000000000000000000 =
50
1
5 (3)
arcsin x arccos2 x
dx
1 .18011660505096216475... 
 
16
x
0
.1797480451435278353...

(2k  1)!!

 (2k )!k (3k  1)
k 1
( 1) k 1
.180267715063095577924...  
k
k  1 k ! 4  ( 2 k  1)

Titchmarsh 14.32.3
2
 2k 
k
. 18028136579451194155...  4 

   
k
2

k  0  k  16 ( k  1)

12



2
 2k  1

1 .18034059901609622604... 
   
k
2 3 
k  0  k  32
  
 4

3
( 1) k k
.1804080208620997292...  2 
 

k
e
k  0 ( k  1)! 2

1
.18064575946380647538... 
10 5  5 
(1  5 ) arccsch 2  5 arcsinh 2 
50
20
5 1
 
 log
32
k 1

(1)
= 
k  1 5k  1

5 .180668317897115748417... 
.18067126259065494279... 
.1809494085014393275...

.18102687827174792616... 

2
16 2  2


2

64
csc 2
3

8
x log x
dx
2
4
0 (1  x )(1  x )

log 3


8 3 24
cosh 1  1

3

 1  2
k 2


1
  (8k  3)
k 1

2


1

2 
(8k  5) 
2


.181097823860873992248... 

e ( 2) 
1
=

k

x
1

dx
8
3
 1  dx
3 4
x
 sinh x
1
2 

 1
GR 4.234.5, Prud. 2.6.8.7

.18121985982797669470... 

 (k  1)  1
2 (k  1)
k
k 1


 1
1 

   k  2 log1  2k   
k 2



3 .18127370927465812677... 
.18132295573711532536... 
2 I1 (2) 

(1)
6 

 2k 
1
 (2k  1)!  k  

k 1

e1/ 6

dx
1 .1813918433423787507...  
x!
1
1 .18136041286564598031... 

1
1 1 4

21 / 3 2 F1  , , ,  1   k
3 3 3
 k  0 2 (3k  1)
1
e ( 5 5 ) / 2 2  e 5 3  5  1  5 
4 .18156333442222918673... 
5e 1  5
1 .18143677605944287322... 


1 .18156494901025691257... 



 1  p 
  F kF!
3

k
k 1
k 1
Berndt 5.28
p prime
=
 (3)   (k )

 (6) k 1 k 3

q 3
2 

3

=
Titchmarsh 1.2.7
q squarefree
1 .18163590060367735153... 
.18174808646696599422... 
2
12
 
=
2
k 1
k
log 2 2
 4
2
1
k ( 2 k  1)
2

3 log 3
3
5
 

  
2
2
2 3
 3
2

25
5
cot 
.181780776652076229520...  125 
 25 log10   (3) 
2
5
6
.1817655842134115276...

GR 3.852.3
 4 2 arcsinh 1  2 log 2 


sin 2 ( x 2 )
0 x 4 dx


 


 
=
.181783302972344801352... 

2 
4
2

log sin
log sin  cos

5 
5
5
5


 (k  3)
1
  (1) k 1

4
3
5k
k 1 5k  k
  50  cos
2
18

3 log 2 2 16



4
9
27

 4k
k 1
3
1
 3k 2
.18181818181818181818
=
2
11
2

3 1
( 1) k
6 
 6 log 2  6 3 log
  3
2
12
3 1
k 1 6 k  k

( k )
.18199538670263388783...   k
k 1 3

1
 1   1 3  1 
1 .18211814116655673118...  HypPFQ  ,11
, ,  , ,   

 2   4 4  16 
k 0  4k 
 
 2k 
.1819778713379401005...


log 2 2
 ( k  1)
  ( 1) k 1
12
2
k
k 1
k



(3 )
1
1
.182153502605238066761...   k
 k k
 k 
3
k 2 3  1
k 1 3 (3  1)
k 2
.18214511576348082181... 
.182321556793954626212... 

 
=
2
  log 2 

1
6
1
1

 Li1     ,1, 0    k 
5
6
6

k 1 6 k

1
2 arctanh 11  2 2 k 1
(2k  1)
k  0 11
log
(1) k 1 k

k 1  2k  k
 2
k 

.182377638546170138737... 
3 1
1

2 F1  2, 2, ,   
2 8
4

.182482081626744776419... 
11 
log 2
 
 3  (1)  3  (2 ) 
6 4
2
(1) k 1
.182487830587610500802...   3
k 1 k  3

1 ( 2)
 (2  i)   ( 2) (2  i ) 
4
1
=  (3)  1   (3, 2  2i )   (3, 2  2i )  
2

2
2
x cos x
dx
=  x x
e (e  1)
0
.18254472613276952213... 

 

 
.182770451872025159608... 
 ( 3)  1 
2

54
27 .182818284590452353603...  10 e 
 (2)
9

1
 

2
k 1 (3k )

log x
4
x
x
1

.18316463254842946724... 
1
log(1  2 cos1) 
4
J248

sin 2 k cos k

k
k 1


k 2
 (k )  1
k3
.183203987462730633284... 
5

 7 / 4  cot 21/ 4   coth 21/ 4   
2 2

k
k 2
4
2

2

=
 2  (4k )  1
k
k 1
.18331367237300160091... 


1  ( 3)  3 
1
     ( 3)    2  3  8 2 log 2  96 G  36 (3)  
512 
4
4


1

=
arctan 4 x
0 x 2 dx
( 1) k  ( k )  1
.18334824604139334840...  

2k  1
k 2

.183392750954659037507... 
1 .18345229451243828094... 

 .1835034190722739673...
(1) k

k 0 k  

 2 4


k 0
( 1) k
2k  1
dx
log cot x

0
2 .183488127407812202908... 
k 2
   1 1
 1 3 
  ,     ,   
2   2 4
1
1

arctan
k
k
1
1     1
  
 
      
2  2 
 2 
 /4
=

  k 
GR 3.511
2

G
 1 
2

3
( 1) k 1 ( 2 k  1)!!
( 2 k )! 2 k
k 1


.183547497347006560022... 
(1) k (k )

2k
k 2
.18356105041486997256... 
coth  1


2



.183578490978106856292... 

 ( 2k )
2k  1

( k )

2k
k 1 2
k 1
.18359374976716935640... 
.18373345259830798076... 

sin  x
dx 
x
x
1
 e e
0

 
k 1


1
 3


23 .183906551043041255504...  e  e
 2 cosh 
488 .183816543814241755771... 
 ( 3)  
 ( 2k )

1 (1) k  (k )
 
4k  1
2
2k  1
.1839397205857211608...


1
k


2 e k 1 (2k  1)!


 
=
x dx
e
x2

1


 
=
.183949745276838704899... 
1 .184000000000000000000 =

 1  dx 1
 1  dx
1 sinh x 2  x5  2 1 sinh x  x3
 5
20
tanh
 5
2

148

125

3 .184009470267851952895... 
1
k
3/ 2
k 2
.1840341753914914215...

k
k 2
.18430873967674565649... 
 .18432034259551909517... 
  (k / 2)  1
k 3
 1 

log
3 
p prime
1 p 


k
log16
 3
4
2
5

3

4
1
x
dx
 4 x
3 log 3

2 
4

( k  12 )!
1 )!( 6 k  1)
2
 (k 
k 1
k 1
2 log 2 5
(1)

 2

3
18
1 k  3k
(1) k


k  0 ( k  1)( k  4)


 1  dx
2
0 x log(1  x) dx  1 log1  x  x 4
8 (3)
4 3

.184471213280670619168... 

9
81 3


2
 (1)1 / 3 Li3 ((1)1 / 3 )  Li3 ((1) 2 / 3 )  
 
=  
1/ 3 
 1  (1) 

1

HW Sec. 17.7
Titchmarsh 1.1.9
1
=
k 1



 2k 1) / 2 1
 k  1
k 1
.18424139854155154160... 


(k )
3
log k

3 .18416722563191043326... 
k
log  (3) 

=
k 1
1
 4k  6
2
Fk k
k
k 1 4

1


3/ 2
1/ 2
k

k
k 2

 4k
2

2 .184009470267851952895... 

1

6
 
=
x log 2 x dx
0 x 2  x  1 
.18453467186138662977... 
1 .18459307293865315132... 
3 2 20


32
27
4 k 2 (4 k 2  1)

2
3
k  2 ( 4 k  1)


k!
1
1 


e1 / 4  erf    
k
 2  k  0 (2k  1)! k  0 (2k  1)!!2
1
=
k 2 ( (2 k )  1)


4k
k 1

ex
e
x2
dx
0
(1) k 1 22 k 1 ( (2k )  1)
1 .184626477442252628916...  

(2k  1)!
k 1
.184713841175145145685...   (3)   (6) 


1
2
 k sin k
k 2

1 .184860269128474...

 1 ( 6) 
 k 1  e
(1) k 1 ( (2k  1)  1)


k!
k 1

.184904842471782293011... 
.184914711864907675754... 
1
2
 
2
e
e 1  1 / e 
8 
8
 1  log  
49 
7
1
  ( 3 / 2 )
.185066349816248591534... 
2 .18528545178748245...

.18533014860121959977... 
1
k 2
1 / k 2



k
x
  x dx

k
e
k 1 e
1

kHk
k
k 1 8

 1  dx
3 3
x
 sin x
1
8 .1853527718724499700...

67


ek H k

k!
k 1
2
 log 2 log 3 2 1
1
 1
.18544423087144965373...  
Li3 (  4) 


Li3    
16
48
12
16  4 
24 .18541655363135590105... 
ee 1    Ei (e)  


 
=

1
log 2 x
x3  4 x

1 .18559746638186798649... 
2
1 .1856670077645066173...


2 k 1

2k


k
k 1 4  1
1
256103
 H ( 3) 5
261000
k 1
1 .185662037037037037037... 
1

32
csc
2


1
1   (k )



k
k
4 k 1 2 k
k 1 2  2



k2
2  2 sin  2  
2
2
2
k  1 ( 2 k  1)
.185784535800659241215... 
.18595599358672051939... 
G  log 2


2
8
 /4
 x tan x dx

 (2k  1)  1
k 1
k
 (1)k 1
GR 3.747.6
0



1
1
2 

 k log1  k
k 2

1
H
2 .186048141120867362993... 
log 2  1  log(3  2 2 )   21k1
2
k 1 2
k


3 ( (k )  1)
4 .18605292651171959669...  
  e3 / k  1 
k!
k 2
k 2
2

1
Hk

.186055863117145362029... 

 8  8 log 2  4 log 2 2  
6
3
k 1 k ( k  1)( 2k  3)


.18618296110159529376... 
(1) k

k
k 0 2  2
.18620063576578214941... 
2

 
=

3
 log 1  x
  2k 

k 1
1 x
1

 3
3
1

  (2 k  1) k
k
dx
0
.186232282520322204432... 
sinh 

2 3

.1864467513294867736...

 4k

 (4k  3)!
k 0
(2)
H k
k
k 1 6 k

2 3
1 

  log 6  2arcsinh

4
8
1 3
2
1
x
=  arcsin x
dx
1  2x 2
0

 ( 3k )
.18646310636181362051...   
4k
k 1

2 I 2 (2)
k 1  2k  k
 


(
1
)
.18647806660946676075... 

e2
k 1
 k  k!


1
  ( 1) k 1  (3k )  1
.18650334233862388596...   3
k 2 k  1
k 1
.186454141432592057975... 
=


3

1 .18650334233862388596... 
log

GR 4.521.3
1 i 3
1 i 3
5 1
  1 i 3  

  1  i 3  
6 6
 2 
 2  


1
 1

5
4
3
2
k 0 k  k  k  k  k  1



 ( 1)  (3k )  1
k 1
k 1
=
=
=


1
k
 3
2 k 2 k  1

3

 1  i 3 
1 i 3 
1

  1 i 3  
  1  i 3  
 2 

2 6 
2











1
 3  2  2 1  ( 1) 2 / 3   2  ( 1)1/ 3   2( 1) 2 / 3 2  ( 1) 2 / 3 
6

1 .1866007335148928206...

.1866931403397504774...

k

1
k 1
1
1
2
1
1

1  1F1  , 3,  4     2 I 0 (2)  2 I1 (2) 
2e
2 3e
2
2

e
k



 
=
 (1)
k 1
k 1
.186744429183676803404... 
.186775363945712090196... 
1 .18682733772005388216... 
1 .18683727525826858588... 
1  2k 
 
(k  2)! k 
1
1
sin 
4
2 cos(1 / 2)

6 3

7

log 2

6
tanh

 7
2


x
5
1
(1) k (2k )! 2k  1
sin

2 k
2
k  0 ( k!) 4

dx
 x 1

k
k 1
2
1
k 2
  (k 11)!3
3 3 3 1 

k 0
k


1
6 (3)   2  6 3  27 
.186945348380258025266... 
16

 2k  k
2 3
  (1) k   k
7 7
k 0
 k 3

 (k  1)  1
.18716578302492737369...   k
k  2 2 ( ( k )  1)
1
12
1
1 .18723136493137661467... 


11 arcsin
11 121
2 3
(1) k 1

3
k 1 k (3k  2)

.18704390591656489823... 
1 .18741041172372594879... 

7


1
k 1 k  2 k 
3  
k 

K Ex. 108d
.187426422828231080265... 
 2 log 2

24
3 (3)
 
16
 /2
x
2
log sin x dx
0

.187500000000000000000 =

3

16
 (1)
k 1
k 1
k
3k
(1) k 1

k 1 ( k  2)( k  2)

=
GR 0.237.5
k 2
 (k )4k

=
 16
k 1
k
1
i i 2
i
1 .187538169020838240502...  1 

 log 2  Li2 ( e2i )    log( x cos x ) dx 
2 24
2
0
1
 (k )  1

.188016647161420393823... 
 (k  2)(k  1) 
k 2
1
9
.18806319451591876232... 
1 .188163855465169281367... 
 log 2
8

G 
log(1  x)
dx 
1  x2
1

 /4

 
=
 log(1  cot x) dx
GR 4.227.13
0

1 .18834174768091853673... 
 (1)k
k 2
.1883873799298847433...

 

=
=
 (k )
(k  1)!

1
1 

1/ k 

  k  ke

k 1


1
1

log 1  2 
2 2 2
1
1 
1
1
2  2 arcsinh 1   1 
arctanh
 
2
4
2
2

1


k
k  1 2 ( 2 k  1)( 2 k  1)


(1) k 2  4k B2 k
1 .18839510577812121626...  csc1  

(2k )!
k 0

11   2

.188399266485106896861... 
6

1
.18842230702002753786...   4
k  2 k  10

k
 (k  4)(k  2)
k 1
2
[Ramanujan] Berndt Ch. 5
.18872300160567799280... 
=
=
2 log(1  2 )  4 log 2  8 
cot

2
3


 4 log 2  2  log sin  log sin
8
2
8
8


 (k  1)
1
  (1) k 1

2
8k
k 1 8k  k
8

 (1  2 )



Prud. 5.1.5.21
1

 
  log(1  x 6 ) dx
=
0

4
16
( k  1) Hk
 1  log   
9
9
4k
k 1
1
.1887703343907271768... 

2(1  e )
4
4 .188790204786390984617... 
, volume of the unit sphere
3

sinh  2
2

3 .1889178875489624223... 
  1  2 
k 
3 2
k 2 
.1887270467095280645...


1
1
 
2
k 1 k!( k  2)!

2
1
.189069783783671236044...  1  2 log   k
3
k 1 3 k ( k  1)
.188948447698738204055... 
I 2 ( 2) 
J149
(1) k 1

k
k 1 2 ( k  1)

=

(1) k 

1



4 k  3
k 0 
.18930064124495395454...   (i )   ( i )   (1  i )   (1  i )

2 
1 .18920711500272106672... 
4
1 .18930064124495395454... 
 (2  i )   (2  i)
.1893037484509927148...

1


1
4


3
 (1)k
k 2

.1893415020203205152..


(1)
sin  x dx
(k  2)!
2
 erf 1
4
2
 (k )
k 1
k 1
.189472345820492351902... 
x
0

1 .18934087784832795825... 
J147



e
1 / k
k 1
HkHk
k
1

2e

x e
1
2  x2
dx
 2 1
 2  
k
k

 
(1) k 1  2k 
  

k
k 1 2 ( 2k  1)  k 

2 arcsinh 2

2

Hk
=  k
2
k 1 2 ( k  1)
.189503010523246254898... 
1

2 .18950347926669754176... 
 k  (k )  1
2
2
k 2
.18950600846025541144... 
.189727372555663311541... 
 (3) log 3 2
4

3
2
log 2 x
1 x 2  x dx 
1    k (2k  1) 
8k
1 
1  

 1 

   1 
   
k
2
2
4
2   k 1
8 2  2 2

k 1 (8k  1)

406 .189869040564760332...

log 2 (1  x)

dx 
x( x  1)
0
1


2
1 e2
e
e  ee 
4
2

cosh 2 k


k!
k 0


 csch 
( 1) k 1
2
2
k 1 k   1

17

(1) k  1
.189957907718062725272... 
csc  5 
  2
20
2 5
k 3 k  5


 ( 2k )
2
1 

  k
   log1  2 
.18995863340718094647...  log
3 3
 9k 
k 1 9 k
k 1
k 1


(1)
 (k )   (k  1)    k  1 log1  1 
.190028484997676054144...  
k
 k
k 2
k 2 k
.1898789722210629262...

1

2
3 1

3 1

1
1 .19020822799902279358...  
k
k 2 1  3
.190086499075236586883... 
1 .190291666666666666666 =
.19042323918287231446... 
log



1
 3 (4k  2)
k 1
k
28567
 H ( 3) 6
24000
7
 4 2 E  1 
2

(k  1 2 )!(k  12 )!
 (k  1)!(k  1)!2
k 1

.190598923241496942...

 2k 
15  8 3 
1
 k
 
6
k  1 6 ( k  2)  k 
(1) k
 (k )  1

k
k 2 2  1
7
1 .19063934875899894829...   
 3

.190632130643561206769... 
k

.1906471826647054863...

1

arctan

2 tan 1
2
1
sin 2 x
 
dx
1  sin 2 x
0

(1) k 

1


k

22 
k 0 

 Ei (1)
9
1
e 
 
.1906692472681567121... 

4
2
2
k 1 ( k  2)!k

1
 
= 
k 3 ( k  1)!  3k !
1 2
 1 
 log 5  2 log 5 log 6  log 2 6  2 Li2    
.190800137777535619037... 
2
 6 

1 .190661478044776833140... 

.190856278271211720658... 
Hk
k
k
k 1
k 1
(1)
k 2
k 1 5 k
i  ( 1)  1  i 
 2  i
 2  i
 1  i
   ( 1) 
   (1 ) 
   ( 1) 
 
 
 2 
 2 
 2 
 2 
16 

=

6

(1) k 1 k
 
=  2
2
k  1 ( k  1)
3
.190985931710274402923... 
5


 1 1
 (k )  1
 Li2    
.1910642658527378865...  


2
k
k k
k 2
k 2 

1
 tanh  1
.19123512945396982066... 
   2
8
5
k 1 4 k  4k  5
k 1

(1)


.1913909914437681436...


k 1
3 .191538243211461423520... 
.19166717873220686446... 
.19178804830118728496... 
3 .19184376659888928138... 
.191899085506264827981... 
1 .19190510063271858384... 
.192046874791755813246... 
22
k
8

2

(2 k )
 k 
5
k 1
2
4 log 2 2 log 3
dx

  4
3
3
x x
1
(2)  1
( 3)  1


1
1
k 1  ( 2k  1)

(
1
)

  sin 


(2k  1)!
k
k 1
k 1  k
2
G
e
e
e 1
J152

.19209280551924242045... 
log k
k
k 1 e
1
sinh  csch  2 
2

.192127760367036391091... 
.1923076923076923076

=
5

26
=
1

2 cosh log 5
.192315516821184589663... 
.192405221633844286869... 
k2 1

2
k 0 k  2

sin 5 x
dx
ex
0


 (1) e
k
 (log 5 ) /( 2 k 1)
J943
k 0
3

3

3
H
.192406280693940317174...   kk 
k 1 8

1
(1) k k  2k 
 
.19245008972987525484... 

k
3 3 k  0 2  k 

(2 k  1)!!

k
k  1 ( 2 k )! 3 k


.19247498022682262243... 
2 log 6  2 log 3  6  
.192547577875765302901... 
csc 2 1
2  sin 2 
8
1 .192549103088454622134... 


k (2k )
k 1
3
26

.192694724646388148682... 
 2k
 1  2e Ei ( 1) 
x
 e ( x  1)
x
2
dx
0
.192901316796912429363... 

 

1 .193057650962194448202... 

 3   log 
  log    

 4
4
4

3

sinh

3
(1) k 1 log( 2 k  1)


2k  1
k 0

[Ramanujan] Berndt Ch. 8



1 

2 

 1  9k
k 1




1
1
  2

2
k  2 4k  2k

1
= 
k 1 ( 2k  1) 2k ( 2k  1)

1
= 

3
k  1 8k  2 k


(1) k
(1) k 1
  5
= 
k 0 k  3
k 1 4k  k
 .193147180559945309417... 
 
 
log 2 

1
 2k (2k  1) 
k 2
J236
[Ramanujan] Berndt Ch2, 0.1
K Ex. 107f

=
1
k
k
2
k 2

=
1
 2 (k  1)(k  2)(k  3)
k 0


 
=

 (k )  1
2
k 2
k

 
=
dx
2 x3  x 2 


 
=
1

=

x
1
dx

x ( x  1) 2



 (2k  1)
22 k 1
k 1


J279
k
4
dx
 x3


1  dx

2
 x3
 log1  x
1
x dx
 (1  x ) sinh  x
GR 3.522
2
0

=
dx
 e e
2x
x
0
1 .193207118561710398445... 
2 .193245422464301915297... 
2 .19328005073801545656... 
 1
9822481
 H ( 3) 7
8232000

2 2
3k
 
9
k 1  2k  2
 k
k
e / 4  i  i / 2 

.193497800269717788082... 
.193509206599196935107... 
G
4
6

3

1 .193546496554454968807... 
7 (3)
 
16



k 1
0
x1 / 3
0
1
k
 x log tan x dx
dx
 e
 k!(2
 /4
 1)

 e

k 1
1 / 2k

1

ek
41 .193555674716123563188...  e
 
k 1 ( k  1)!
.19370762577767145856...   (3)   (7)


3
1
k 1  (3k )  1
   log e  k  1
.193710100392006734616...   ( 1)
k!
k 1
k 2 k
e 1

.193998857662600941460... 

1 i 3
1 i 3
  log  

 log 2  log  
 2 
 2 


1

log 
1/ 3
2/3 
 2   (1)  (1)



1

k  1  ( 3k )  1
=  log 1  3    ( 1)


k 
k
k 2
k 1

=
 


.194035667163966533285... 
1
6
3

csch 

6
3
2
(1) k

2
k  0 2k  3
.19403879384913797074... 
( 1) k  ( k )  1


k2 1
k 2
1 k2
1 1 1 

log 1     



k  2 4k 
k 2  2k


.194169602671133044121... 
 (1)k 1
k 1
.194173022150715234759... 
.194182706187011851678... 
.194444444444444444444 =
.19449226482417135531... 
2 .194528049465325113615... 
3 .194528049465325113615... 
4 .194528049465325113615... 
57 .194577266401621023664... 
 
 

1
i
 1  (1)1 / 4   1  (1)1 / 4  
4
2
i
  1  (1)3 / 4   1  (1)3 / 4
4


k

(1) k 1  (4k  1)  1


4
k 2 k  1
k 1
 


1 cot 2
1


  2 2
4
2 2
k 1 k   2


k
1
7
1

  k    ,  1, 0   
36
7

k 1 7
k 1 k ( k  1)( k  3)
1
 2

e2  3
2k k
 
2
k  1 ( k  2)!
2

e 1
2k
 
2
k  0 ( k  1)!
2

e 1
2k k
 
2
k  1 ( k  1)!

 2k  1
2 e2  I 0 ( 2 )  I1 ( 2 )    

k  1  k  ( k  1)!
1

4e1/ 4
5 .194940984113795745459...  9
.194700195767851217061... 
1 .194957661910227628163... 

H ( 3) k
4k
 
=


2
erfi

 (1)
k 1
k 1
1

2
k
k !4 k

 k !2
k 0
k
1
( 2 k  1)
.195090322016128267848... 
.195262145875634983733... 
.195289701596312945328... 
7
16
16
 /4
2 2
sin 3 x
 
dx
4
3
0 cos x
 cos

 ( 2) 
3
cot

k2 1


4
2
k  2 3k  k

1 
3

 (2k )   (2k  1)
k 1
3k

 2k 

.19541428041665277483... 

sin
  k   (4k )  1 

k 1


.19545006939687776935... 
 x (2 x  1)  1 dx
1

Bk k
k 1 ( k  1)!
.195527927242921852033... 

.195686394433340358651... 
2 
1 .195726097034987200905... 


2
 2 HypPFQ{1,1,1},{2,2,2},1 
6
cot


log 2 x
1 e x dx
2
2
 2 log 3 log 3 3 1
1
 1
.19573254621533971315...  
Li3 (  3) 


Li3    
12
72
72
12  3


 
=
log 2 x
dx
x 3  3x

1

.19582440721127770263.... 
 (1)
k
 (k )  1) 2
k
 sin 1
(cos 1)/ 2
1 .195901614544554380728...  e
cos

 2 
i
1 ei / 2
=
e
 ee / 2
2

1
2 .196152422706631880582...  3 3  1   k
k 0 6
k 2


5 .196152422706631880582... 


k 0


k
 2k 
  
k 
27
.19621530110394897370.... 


1
csch


2 2 3
3
.196218053911705438694... 
9 log 3 3 log 2


8
2
1 .19630930268377512457... 
cos k
 k !2

2

tanh 1 
2x
 sin 
0
(1) k 1

2
k 1 3k  1


H 2k
k
k 1 9


dx
sinh x
GR 1.463.1
.19634954084936207740... 

16

(1) k

4
k  0 ( 2k  1) ( 2k  1)  4

( 1) k 1
2k  1
sin

2
4
k 1 ( 2 k  1)
3


sin k cos k
sin 3 k cos2 k



k
k
k 1
k 1


2
2
x dx
x dx
0 ( x 2  1)3   ( x 2  4)2

=
=
=
=


=
e
1 .1964255215679256229...

K Ex. 107e
J523
dx
 e 4 x
4x
0


H
2k
( ( 2 k )  1)
k 1
1 .1964612764365364055...

4G 
 /2
=

0

2
4
1
arcsin 2 x dx
0 x 2

2
x cos x
dx
sin 2 x
GR 3.837.5
arctan 3 x
x
dx
1  x2

 1 

1  2k 
   HypPFQ  ,{1,1}, 4  1
4 .19650915062661921078...  
2 
k 1 ( k!)  k 
 2 

3
13 (3)
2
log 3  (1)  1 
 



.196516308968177764468... 
   
9
6 
27 3  18 3
 3

Hk
= 
2
k 1 (3k  1)
=
2
0
.196548095270468200041... 
.196611933241481852537... 
 
e

( e  1) 2


.1967948799868928...

 ( 1)
k 1
(k )
5k  1
2k
k!2
(1) k 1 k


ek
k 1

(1) k

k  2 k ( 2k  1) log k


k 1
k 0
3

2
1 .19682684120429803382... 
.1968464433591154...

 J 0 2 2   (1) k 1
GR 4.534

3 ( 3)
.196914645260608571900...  2 

2


 
=
.196939316767279558589... 
x2
0 e x e x  1) dx 

x 2 dx
0 e3 x  e2 x
dx
1 x3  x 2 

9
( 1) k

k  2 k ! ( k )

.196963839431033284292... 
25 cosh 1 27 sinh 1 13e
1




32
32
16 32 e
2 .19722457733621938279...  2 log 3
2 .19710775308636668118... 

k5

k  1 ( 2 k )!
( 1) k 1 4 k

k 1 ( 2 k  1)!( 2 k  1)
 5
2
1 .19732915450711073927...    8G  16    2, 
 4

1
2
(1)  1 
17 .19732915450711073927...    8G      

2
 4
k 1 ( k  1 / 4)
.19729351159865257571...  1 
.19739555984988075837... 

1 
(1) k

arctan   2 k 1
5 k  0 5 (2k  1)


k 1



1
(1) k
.197395559849880758370...  arctan   2 k 1
5
(2k  1)
k 0 5


si ( 2)

2
 
=
1 .19746703342411321823... 
.19749121692001552262... 
1 .197629012645252763574... 
.197700105960963691568... 
.197786228974959953484... 
.197799773901004822074... 

1
 arctan 2(k  2)
3 2


8 12
2
[Ramanujan] Berndt Ch. 2, Eq. 7.6

 k ( (2k )   (2k  1)  2) 
k 1
3 sin 1 
k2
2  cos 1 
  ( 1) k 1
2
( 2 k )!( k  1)
k 1

2


coth  1  
2
4
4
k 1 16k  1



 ( 2k )
1
1

 
  k
2 6 3
9
k 1 (3k  1)(3k  1)
k 1

1
kH
4  3 arccot 2  2 log 5  4 log 2   (1) k 1 k2 k

25
4
k 1

k
60  22e  

k  1 k !( 2 k  10)
.19783868142621756223... 
2
4


 log 2
log 2 2  2
log x sin 2 x
dx


 
x
4
48
0
2
13 (3) 1 ( 2)  7 
5 3 39 (3)
19 3
 27 
    


8
16
4
6 3
12 3
6

(1) k 1
= 

3
k 1 ( k  1 / 3)
3 .19784701747655329989... 

 

.19809855862344313982... 
1

3
2 
k 2 k  k

 ( 1)  (5k  2)  1
k 1
k 1

.19812624288563685333... 
1 .19814023473559220744... 
log k
3
k 1 k
  ( 3)   


 
=

0
5
   1   
2  4  4
 /2
sin 2 x
1  sin 2 x
dx 

sin x dx
0

.198286366972179591534... 
  3
2 E (1)  K (1)  
  (2k )   (2k  1)
2
k 1
.198457336201944398935... 
H1 (1) 
(1) k

 k  0 ((2k  1)!!) 2 (2k  3)
2

( (k  1)  1) 2
2k
k 1

3e 4  1
4k k
 
4
k 1 ( k  1)!

.198533563970020508153... 
41 .198612524858179308583... 
 (1)k 1
.198751655234615030482... 
(1) k

k
k 0 3  2
.198773136847982861838... 



csc 2 3 
4 3
( 1) k
.19912432950340708603...  
k  0 k ! k !!
17

66
(1) k

2
k  4 k  12


1
.199240645990717318999... 
3  2  2e  2 Ei (1)    e x x 2 log x dx
0

.199270073726951610183... 
 ( 1)
k 1

.199376013451697954050... 
k 1
log k
  2k 
k 1
 
k 
(k )
k
log (2 k  1)
 log 8
40



147 14
7
1
.19947114020071633897... 
8
.199445241410995064698... 
1 .19948511645445344506... 
2 .199500340589232933513... 
.199510494297091923445... 

 
=
1
 k ( 4k  7 )
k 1
2
1
 2
 log 2 log 3  log 2 3  Li2    Li2  
6
 3
 3

2k
1
 k
Berndt ch. 31
 2 tan k  

2 
k 0 
1
1

cos 2  CosIntegral (2) sin 2  SinIntegral (1) cos1 
4
2
2

dx
0 e x  x 2  4
2

sin 2
(1) k 1 k

3
k 1 k  2
x
x
 root of ( x  1) e  ( x  1) e

.199577494388453983241... 
1 .199678640257733...

5 .199788433763263639553... 
 3 
.20000000000000000000
=
1
1
 sin 2 arctan 
5
7

=
=
.20007862897291959697... 
=
.2000937253541084694...

 
3 .2000937253541084694...
.2002462199990232558...

1
 (k )



k
k
k 1 6
k 1 5  1

cos 2 xdx
log 4 xdx

0 e x
1 x

k ( k  5)

k 1 ( k  4 )( k  1)
e
J1061


1
 (2  i)   (2  i)   (1  i 3 )  4  i  3 
6
2


4i 3
4i 3
  (1  i 3 ) 

 (1  i 3 ) 



2
2




4i 3

 (1  i 3 ) 

2


3


k
  ( 1) k 1 ( ( 6k  3)  1)

6
k 2 k  1
k 1

( 1) k 1


2
3
k 1 3k  k

1
1  dx

3
=  log(1  x ) dx   log1  3  2
x x

0
1



 2 log 2  3 


1
1
 2 log 2         
3
 3
6
2  2 2 log(1  2 )  log 2 
1
0

2
k 1

 
.2003428171932657142...
=

1 .20038385709894438665... 

1
dx
3

 log1  x
k
1

( 2 k  1) k
2  2 2 arcsinh1  log 2 
(3)
6
2 2  2 log 3 log
729
 1
 2
 8Li2     12 Li2   
64
 2
 3
=
(1) k 1


k  3 StirlingS1( k ,3)
1 .2004217548761414261...

 1
 1
1
     
 ¼  ! ¾  !
 ¼
 ¾
.2006138946204895637...

1  log 2 1  i  
i
i

3 i
 3 i 

  i  1      1         i    


 2 2




2
8 
2
2
2 2


 
=
=

1  1 i 
i
i
1 i


            1      1     tanh 
 2 2
 2
 2
8   2 2
2

3 log 2 
 
 coth
4
2
8
2

( 1)k 1

3
2
k 1 k  k  k  1
1
.2006514648222101678...

Ei(  e)  Ei( 1) 
e
e x
dx
0
.201224008552110501897... 
6 .201255336059964035095... 
.2013679876336687216...
sin 2  sin 2 1 
(1) k 4k


k  0 ( 2k )!( k  1)

3
5
9  e2

8


2k k

k  0 ( k  3)!
(( k )  1)2
k!
k 2
1
6
3


csch 

6 3
2

.20143689688535348132... 
.2015139603997051899...
.201657524110619408067... 
 (1)k


3
k 1
.201811276083898057964... 

 
=
.20183043811808978382... 
.201948227658012869935... 
.20196906525816894893... 

 
=
1 .20202249176161131715... 
2
cos x  e  x
   
dx
2
2
x
0

1 .20172660598984245326... 
( 1) k


2
k 0 2 k  3

2
k
1
2
 1
2
 3 log 2 3  3 log 2 log 3  2 Li 2     3Li2   
2
 2
3
k 1

(1) H k

k
k 1 2 k ( k  1)

e 3
k2

 
8 8e
k 1 ( 2 k  1)!

e sin 1
sin k
  (1) k 1 k 
2
1  e  2e cos1
e
k 1
i
i 
 

16  4 (3)   (5)  8  1      1    
 2
  2


1
 ( 2 k  5)
  ( 1) k 1

7
5
4k
k 1 4 k  k
k 1
1
2G  1
1

.2020569031595942854...

1
1 x log 2 x
 (3)  1   3  
dx
2 0 1 x
k 2 k
GR 4.26.12

=
6 x  2 x 3 dx
0 (1  x 2 ) 3 e2x  1
1 1 1

 
=
Henrici, p. 274
xyz
   1  xyz dx dy dz
0 0 0

1 .2020569031595942854...

=
=
=
1

3
k 1 k
MHS (2,1)

Hk

2
k 1 ( k + 1)

H k (2k  1)

2
2
k 1 k ( k  1)
 (3)  
3k 2  3k  1

2
3
k 1 k ( k  1)

Berndt 9.9.4
H ( 2) k

k 1 k ( k  1)

=
=
5  ( 1) k 1
5  ( 1) k 1 ( k !) 2



2 k 1  2 k  3
2 k 1 (2 k )! k 3
 k
k
Croft/Guy F17
Originally Apéry, Asterisque, Soc. Math. de France 61 (1979) 11-13
( k !)10 (205k 2  250k  77)
64((2 k  1)!) 5
=
 ( 1) k 1
=


7 3
1
7 3
 (k )
 2 3 2k

 2  32k
 1
180
180
k 1 k e
k 1 e
Elec. J. Comb. 4(2) (1997)
Grosswald
Nachr. der Akad. Wiss. Göttingen, Math. Phys. Kl. II (1970) 9-13
=

 (k ) 
1
2 3
 4  32k  2 2 k 2  k  
2
45

k 1 e
Terras
Acta Arith. XXIX (1976) 181-189
=
4 2
7
 (2 k )
2 2
2
log 2 


k
7
7
k  0 4  2 k  2
=
4 2   (2 k )
2 2
2 2
log
2



9 k  0 4 k  2 k  3
9
27
=
 2 2 

 (2 k )

k 0
4 2 k  2)(2 k  3
k
Yue 1993
Yue 1993
Yue 1993
=
=
=
=
4 2 
 (2k )

k
7 k  0 4 ( 2 k  1)( 2 k  2 )
8 2 
 (2 k )


k
9 k  0 4 (2 k  1)(2 k  3)


2
3
Ewell, AMM 97 (1990) 209-210
Yue 1993
(2k  5) (2k )

 4 (2k  1)(2k  2)(2k  3)
k 0
k
Cvijović 1997
4 2 
3  2 
(2k  5)(1   (2k ))
1  3 log  

k
3 
2  3 k  0 4 (2k  1)(2k  2)(2k  3)
Cvijović 1997
=
2 3
(2 2 k 1  1) (2k ) 
  log  4 k


14  2
k  0 16 k ( 2k  1)( 2k  2) 

4
Ewell, Rocky Mtn. J. Math. 25, 3(1995) 1003-1012
=
 (6)
 1  p 
4
p prime
=
=
=
1  p 1  p 2  p 3
 1  p 1  p  2
p prime
1
3
5
HypPFQ[{1,1, 1, 1},{ , 2, 2},  ]
4
2
4
1
2
1 log x
dx
2 0 1  x
 (4)

=


GR 4.26.12

log 2 x
x
1 x 2  x dx   0 x log 1  e dx
1
=
log 2 (1  x)
0 x dx
1
=
 log(1  x) log x
0
=
4
3
GR 4.315.3
 /4
 log(cos 2 x) tan x dx
GR 4.391.4
0
 /4
=
dx
x

0
sec 2 x (log tan x) 2
dx
1  tan x
1
=
16
  arctan x log x (log x  2) dx
3 0
=
1 x 2 dx
1 x 2e x

dx
2 0 e x  1 2 1  e x

0
GR 4.594

=
x 5dx
e
x2
1
0

=
2
e 3 x
dx
x
3  ee  1
=
1
e 3 x
dx
x
2  e e  1
GR 3.333.2

GR 3.333.1
0
=
 Li e dx
x
2

2 .2020569031595942854...

 (3)  1 

 k ( ( k )   ( k  2)) 
k 2
2k 2  k  1

3
k  2 k ( k  1)



1

( (16k  13)  1)

3
 k 13 k 1
k 2


1
.20206072056927833979...   3

( (15k  12)  1) 

14
k 2 k  k
k 1
.20205881140141513388... 

k

.2020626180169407781...

 ( 1)
k 2

.2020645408229235151...

.20207218728189006223... 
.20208749884843252958... 
.20211818111271553567... 
.20217973584362803956... 
.20218183904523934454... 
.20230346757903910324... 
.2025530074563600768...

=
k
log ( k )
k

1

( (14k  11)  1)

3
 k 11 k 1
k 2


1

( (13k  10)  1)


10
3
k 2 k  k
k 1


1

( (12k  9)  1) 


9
3
k 2 k  k
k 1


1

( (11k  8)  1) 


8
3
k 2 k  k
k 1


1

( (10k  7)  1)


7
3
k 2 k  k
k 1
k


2

k 2  (k )
( 1)
   e 2 k  1   


k!
k
k 2
k 1


1

( (9k  6)  1)


6
3
k 2 k  k
k 1
 7
i
i
 
 cot (1)1 / 4    cot (1)3 / 4   
4 16 4 2 8
i
1
  (2  i )   (2  i )    (1)1 / 4    (1)3 / 4 
8
4


1
  ( (8 k  5)  1)

3
5
k 2 k  k
k 1
k
1

1  1
.2026055828608337918...    ,4,0   Li4     k 4 
5

 5  k 1 5 k

3 cos1
k3
.20261336470055239403... 
  ( 1) k
( 2 k )!
8
k 1

2
 (2 k )
.20264236728467554289... 
2  
2 

k 1 (2 k )

 
=
1
1
0
0
  x cos x dx    x 2 cos  x dx 


 
=
dx
 e
0
.20273255405408219099... 
=
x1 / 2
log 3  log 2
1
 arctanh 
2
5


1
1
1
  k 

2 k 1
2 k 1 3 k
(2 k  1)
k 0 5
 1  H (n) k
    k

2 k 1 3
n 1 

=

=

H k 1


k
k 1 3
AS 4.5.64, J941, K148
( 1) k 1

k 1
k
k 1 2




log x dx
 (2 x  1)
2
1
1
=

 ( x ) sin  x sin 5 x
x
0
.20278149888418782515... 

GR 1.513.7
dx
i

 2  e 1 / 2    2  e 1 / 2    2  e1 / 2    2  e1 / 2 
4

=
  (2k  1)  1sin k
k 1

.2030591755625688533...
.2030714163944594964...





1
( ( 7 k  4 )  1)



3
4
k 2 k  k
k 1


 ( 6k  3)  1
 
   k 3 log(1  k 6 ) 
k
k 1
k 2

3 .203171468376931...

2 .2032805943696585702...

.2032809514312953715...

.2034004007029468657...

( x)  1
i i  i i
tan(log  )  i
   i
 3
log   
 4
root of
1    Ei ( 1) 
( 1) k 1


2
k 1 k !( k  1)

( 1) k 1


k  2 ( k  1)! k !


.2034098247917199633...


 .2034498410585544626...

(2)  3(3)  2( 4)
 33
( 1) k

k  0 ( k  2 )( 3k  2 )


12

.20351594808527823401... 
 log (3k )
k 1

.2037037037037037037
=
.20378322349200082864... 
2 .20385659643...

.20387066303430163678... 

11

54
k 1
 ( 2 )  2 ( 3) 
 
=

.2040552253015036112...

 4k
6
 
 
=
.20409636872394546218... 

 
=
.204137464513048262896... 
.20420112391462942986... 
.20438826329796590155... 


1
1  2

p ( p  1) 
p prime1 

32  2 2  4

 768 log 2  16 ( 3)  (5) 
3
45

 ( k  5)
  ( 1) k 1
4k
k 1



5
3 5 

   3 5 5  2 5    5 log 5  20  5 log
 4
6
2 
 5k

 (k  2)
1

  (1) k 1
2
k
5k
k 1

3

( 4 )  4( 2)  4  coth  8 
2
1
( 2 k  4)

  ( 1) k 1
4
k
4k
k 1
k 1
 3  i 3 

1 1   3  i 3
 
 

  
3 12 6   2 
 2  


1
( ( 6 k  3)  1)



3
3
k 2 k  k
k 1


 ( 5k  2 )  1
k 2 log(1  k 5 ) 




k
k 1
k 2


 (4 k  1)  1
 1
  k Li2  4 

2
k 
k
k 1
k 2
( 3)

H k

k
k 1 6


3
5
=

k 2
5
k 1
.2040955659954144905...
k
 ( k  2)

1
 k5



1024  128 
k 1
61 .20393695705629065255... 
26

27

 2
p


1
  p 2  1 ( p  1)   6
p prime 



1
 k ( 3k  9 )
 4k

6

2 .204389986991009810573... 
 2 
log

 log 2 
1 .20450727367468051428... 
e 

4
[Ramanujan] Berndt ch. 22

x
2
2
e1 x dx 
0
 k2  2
 
2 sinh  2 csc h    2
k 0  k  1 

5 .204869916193387826218... 

 2 k  ( 1) k 1
2 log 1  2  2 2  2 log 2  3     k

k  1  k  4 k ( k  1)

k
.205074501291913924132...  4 log 2  log     2   k
 (k  1)  1 
k 1 2 ( k  1)
1

1  1
.205324195733310319...
  , 3, 0   Li3     k 3
5

 5  k 1 5 k
k

 cos
2
( 2)
( 1) k 1
2 
.205616758356028305...


 
 
2
2
48
8
4
k
k
k 1
k 1
1
 
=  Li2 (i )  Li2 (i ) 
2

Hk
= 
k 1 k ( k  1)

.20487993766538552923... 




 
=
log x dx
x3  x
1

1

 
=

0


 
=
x log x dx

x2  1

1  dx
4
 x
 log1  x
1

( 1)k 1

k
1
k 1

1

   1  k  
6 
k 1 
.20569925553652392085... 
1 .2057408843136237278...
2

73
coth k 
 

180
k3
k 1

3
3 1  4 
.205904604818110652966... 

e  x sin x 3 dx



5/3

2
 3
0
.2057996486783263402...



2  1
.20602208432520564821...   Li2    
3  3
1
log 2 x
0 ( x  3)2 dx 
Ramanujan
.2061001222524821839...

2
36


.2061649072311420998...

 ( k

49

720
2
k
k 1
2
1
( k  6)
 1)  1 
k 2
.206183195886038352118... 
.206260493108447712133... 
.2062609130638129393...

.20627022192421495449... 
.2063431723054151902...

 

=
.20640000777321563927... 
.20640432108289179934... 
.2064267754028415969...

 
.2064919501357334736...

=

1 .206541445572400796162... 
1 
13
21  8 5   Fk Fk Fk
 6 arc coth

 log
11  k 1 6 k k
5 5 
5

i  (1) 
i
i 
(1) k 1 k

 1     (1) 1     
 (2k  1) 
8
4k
 2
 2 
k 1


1

( (5k  2)  1)   ReLi 2 (i ) 


3
2
k 2 k  k
k 1


 ( 3k  2 )
1



k
5
2
6
k 1
k 1 6 k  k
 1
2  2 log 2 2  4 log 2 log 3  2 log 2 3  4 Li 2  
 3
k

( 1)


k
2
k 0 2 ( k  2)


1
k 1  ( 3k )
(

1
)



k
3
5
k 1
k 1 5k  1


 ( 4 k  1)  1
k log(1  k 4 ) 




k
k 1
k 2

 3

3
1
coth

  2

6
3 4
k  2 3k  1

( 2 k )  1

( 1)k 1

3k
k 1


1
k
H
(

(
k

3
)

1
)

log


k
3
2
k 1
k 1
k 2 k  k
k

1
3
 (k )  1  3
log   (1   )  
Srivastava
  
 2
2
2
k
k 2
J. Math. Anal. & Applics. 134 (1988) 129-140


.2066325441561950286...


 (3k )  1
k
k 1
2


 1
3 

 Li  k
k 2
2
 2k  2k
k
(

1
)
 

 k  k!
k 0

1
1 .20702166335531798225...  1  e  erfi (1)   

k 1 k !( 2 k  1)
.2070019212239866979...

I 0 ( 4)

e4
1

 
=

2
ex  1
0 x 2 dx 
GR 3.466.3
5 .2070620835894383024...

ee  1

e

ek


k  0 ( k  1)!

3 .20718624102621137466... 
4 .2071991610585799989...

 (3)
2
25 2 3023


288
1728


7 (3)

2

H k H k 4
 k ( k  4)
k 1
0
x dx
log2 (1  x )


0 sinh x
 (1  x) sinh(log(1  x)) dx 
1
2
1 
 1 
 
erfi ( e )  e erfi    
 e 
4 e
(5)
.2073855510286739853... 

5

k 1  (2 k )  1
.20747371684548153956...   ( 1)

k (2 k  1)
k 1
1 .207248417124939789...


8
.20749259869231265718... 


10 75
.207546365655439172084... 
.2078795763507619085...


 (
3
k 0
x
4
arcsin xdx 
GR 4.523.1
0
 (k )

 ( 1)
k 2
k
k ( k  1)

(k )   2(k )) 
k 2
i
i  e  / 2 (principal value)  ei log i 
 
 
cos(log i )  i sin(log i )  cosh   sinh  
2
2
 1
.20788622497735456602...       
 2
 
3 .207897324931068989...
1 .208150778890...
.2082601377261734183...
.2082642730272670724...


sinh k
 k !(2k  1) 
1
1
log 2    2 
2

2 .2077177285358922614...

 
.20833333333333333333
=
e e  e 2 1/ e  e 2  1

2e

1
  4

k 2  (k )


3

32 2

cosh k
 ( k  1)! 

k 0
dx
 ( x 2  2)3 
 /2

sin 5 x cos3 x dx 
0
8
  (4)  192 log 2  4 (3)  256 
3


1
( k  4)
= 
  ( 1) k 1
5
4
4k
k 1 4 k  k
k 1

32 
=
5

24
2

log(1  x )
dx 
x5
1

.2084273952291510642...

.20853542049582140151... 
.2087121525220799967...

e( Ei (2)  Ei (1))
1
 Li3  2 
8
5  21

2

0
log(1  x)
dx 
ex
log x
dx
3
 2x
1
7
k 1

2
x

1

 2k 
1
 
k
( k  1)  k 
k 1
 2k 
( 1)
 
k
k  0 3 ( k  1)  k 

1
11/ e
1 .2087325662825996745...  e

e  
k
k  0 ( k  1)! e
i
i
   ( k  3) 


.20873967247130024435...   ( 3)  4  2  1    2  1    Re

k
 2
 2
 k 1 (2i ) 


1
( 2 k  3)
 
= 

( 1) k 1


5
3
4k
k 1 4 k  k
k 1



log 2

e
 
=
1

 I 0 (1)  StruveL0 (1)    esin x dx 

6 .208758035711110196...
0
.2087613945440038371...


2
( 1) k 1


2
k 1 k ( k  1)
   ( k  3) 
  Re

k
 k 1 (2i ) 
 2 log 2  2 
12
( 1) k

3
2
k 2 k  k

=

J365
1

 
=
  log x log(1  x )dx 
0
1 .2087613945440038371...

2 .2087613945440038371...

2
1
 2 log 2  1 
12
2
 2 log 2 
12

H2 k
 k ( k  1)
k 1
1
=
(1  x ) log(1  x )
dx
x
0


  log 1 

0
1
 log x dx
x

=
log(1  x )log x
dx 
x2
1


1 .20888446534786304666... 
 ((k )  1)
k 2
k 2
.208891435466871385503... 
 (1   ) cos1 


 
=
1

log 2  e  i   log 2  ei  
2
cos k
 (k )  1 
k
k 2

GR 4.221.2
6  1 1
1 
1

.20901547528998000945... 
Li2      k
5  6 5
5 k 1 6 ( k  1) 2


1
k  ( 4 k  1)
.2091083022560555467...   ( 1)


5
k
4
k 1
k 1 4 k  k
1 1   3  i
 3  i
=     
 
 
 2 
2 2  2 
.2091146336814109663...

 2
8



2
1

3

 (2 k  1)

.2091867376129094839...
tanh
 (2k  1)!

k 1
.2091995761561452337...
1 .2091995761561452337...


2
1 
3 3
2

3 3

=
  2k 
k 0


 4k
k 1


H (2) k

k
k 1 6
1
 4k  3
2
1
1
  sinh k  k  
k 1
 /2
(1  sin x ) 2
dx 
2  sin x

0

k!


k
k  0 ( 2 k  1)!! 2
1

( k !) 2
CFG D11, J261, J265

k  0 ( 2 k  1)!
  (2 k  1)
k


9k 2
= 

k 1 ( 3k  1)( 3k  1)


dx
dx
  x

=  3
x 1
e  e x  1
0
0
 


 
=
dx
 2  sin x 
0
x dx

3
1
x
0

=

GR 2.145.3
dx

2
x  x 1

0

x
1
2
dx
 x 1
.209200400411942123073...


13  4 5
4
14311  4955 5 
 2 5 arctan h


5
log

log

89
3
5046
5  5 5 




 
1


 ( (k )  1)
k 2
k 2
1 .20930198120402108873... 
1 .20934501087260716028... =
81 9 3 3 2



2
2
2
 2 log 2
4 2

( 1) k 1

2
2
k  1 ( k  1 / 9)

log 2 x dx
 
2 x2  1
0
.20943951023931954923... 
.20947938452318754957... 
.2095238095238095238
=
.209657040241010873498... 
.2099125364431486880...
.2099703838980124359...
2 .2100595293751996419...

15
log 2 

1
( k )
 k 
2 k 2 2 k
Prud. 2.5.29.24


 k
k  2 j 1
1
j
2k
j



 2k 
22
1
1

 k
 
105 k  0 (2 k  1)(2 k  9) k  0 4 ( k  2)( k  4)  k 
log
(  1) csc 


 (2 k )  1
 kk
k 1



72
k2
1

1
  ,  2, 0   Li 2     k 
343
8

 8  k 1 8

k log k
  ( 1)

k!
k 1


10 3

81 3

=

0
1 .2100728623371011609...
(sin x  x cos x) 2

x6


1
log 2 x dx
0 x 2  x  1 


log 2 x dx
x log 2 x dx
0 x3  1  0 x3  1 
GR 4.261.2
log 2 x
dx 
x 2  x 1
2 2
8
13 2
log 2 2
1
1 1 1
 1 
  , 2,  


Li2 

8  2 2
48 2
8 2
2  2
1

 
=
.21007463698428630555... 
log x dx
2
1
0
 2x

 (k )
k 1
k
 (3)
log  (3k ) 

k 1
4   3
1 
 
6 


1
.2103026500837349292...  

k
k 1 (5k )
.21010491829804817083... 
.21036774620197412666... 


 (k )
k3

( k  1)!
k 
1
2 )! 4
 (k 
k 1

sin 1  ( 1) k
( 1) k k 2



4
k  0 4( 2 k  1)!
k 1 ( 2 k  1)!( 2 k  1)
2 .2104061805415171122...

(3)  ( 7) 
.210526315789473684
=
4

19
.2105684264267640696...

6(log 3  log 2) 
8 .2105966657212352544...

2

(3)



20

9

( 1)k


k
k  0 2 ( k  1)( k  4 )
Titchmarsh 1.6.2
Dingle p. 70
.21065725122580698811... 

12


( 1)k


k  0 ( 2 k  1)( 2 k  4 )
log 2 1
 
6
6
1

 
=
x
2
arctan x dx 
0
.21070685294285255947... 

1 1


3 3e
1

1 .21070728769546454499... 
 1  dx
3 7
x
 cosh x
 log(1  e
k 0
k
)  k

( 1) k 
1 


2k 
k 1 

1 .21072413030105918014... 



46 .21079108380376900112... 
.2109329927620049189...

 

 

 
17e
1
 2 (i )  2 ( i )  4  1 
8

 3 1
k
=
   (2  i )   (2  i )    4

2 8 4
k 2 k  1
 1 1
=
   (i )   (i )  
2 8 4


=
 ((4k  1)  1)
k 1

.210957913030417776977... 
.21100377542970477...

 
2 .211047296015498988...
1 .2110560275684595248...

1
(1) k 1 H k




k 2
4k k
k 1 5 k
k 1
 4
2
=  ( 2 )  log 4 log 5  log 5  Li2   
 5

1
1

 ,2, 0   Li 2 ( ) 
5
5


2 log    log 2 

.21107368019578121657... 

7 .2111025509279785862...

1 .21117389623631662052... 
.2113553412158423477...
x2
0 e x ( x  1) 2 dx
2  3e Ei ( 1) 

E(¾) 

 (3k )  1


k!
k 1



( 1) k 1 / k

k 1
 e

1/ k 3

1 
k 2
52  2 13

 1
 1

 ( 3k )

k 1
2k

k 1

( 2 k  1)!!
 (2 k )!! 
k

J166
1
Berndt 7.14.2
(1   ) 
2

 

 (3k )  1
1
.2114662504455634405...  
   3k    log1  m 3  
k
k 1
k  1 m  2 km
m 2
 1

 3
 3
  log 3  log cosh
 
=  log
cosh

2 
2
 3
.2113921675492335697...


5i 3
5i 3
  log 

log 

 2 
2





e
sin ke
.2116554125653740016... 


2
k
k 1
 
.2116629762657094129...


=
 1 
8 .21168165538361560419... 
.211724802568414922304... 

Ei(e)

2
2

4


 
=
coth  
4

3
 3G
3 2 
  7 (3)


3
 log 2  G  2   


32  2
16 
4
4
Hk
 (4k  1)
k 1
2
(1) k !

k  3 k ! 2
16
.21184411114622914200...  8 log 2 

3

.21180346509178047528... 

1


k
k  0 2 ( 2k  8)

1
 2 (k  3)
k 1
k
1
.211845504171653293581... 
1
  log 2  ci (2)    log x sin x cos x dx 
4
0

.2118668325155665206...

.21192220832860146172... 
1

k 1 e k ( k  1)
2  e  ( e  1) log(e  1)  
3
8
( 1) k 1 k

 ( k  1 / 2)
 4G 
k
k 1
3

1
1/ 3
k 0
 0 
2
.21220659078919378103... 
 
 
3
 3 / 2
1 .21218980184258542413... 
3
k
1
 x sin
7 .2123414189575657124...

6 (3) 
e
0
3
x
 x dx 
0
1
.21231792754821907256...  log 3   2 log 2 
2

3
2
x
3
1
x dx

 e x  2
dx
 x2
J149


 
=
e
0
.2123457762393784225...


e
1


.21236601338915846716... 
dx
x1 / 3

 (2 k  1)  1
2


1
 1
2 

 k Li  k
k
k
k 2
H log k
3 .21241741387051884997...   k
k  2 k ( k  1)

( k )k
.21243489082496738756...    k
k 1 2  1



cos 2 x dx
cos x dx
  2
.2125841657938186422... 

2  
2
2e
x 1
x 4

0
k 1

AS 4.3.146, Seaborn Ex. 8.3
 /2
 sin(2 tan x) tan x dx
=
GR 3.716.6
0

=
 cos(2 tan x) sin x
0

.2125900290236663752...

GR 3.881.4
log k
3
1
k
k 1
.2126941666417438848...
dx
x
1 
Hk
 log 2  log 2  

2 k 1 4 k 3  k
2

10  3 2  24 log 2
( 1)k 1
 3
2
108
k 1 k  3k

1
1
.2127399592398526553...  I3 (2)  0 F1 (; 4;1)  

6
k  0 k !( k  3)!

( 1) k 1 (2 k  1)!!

.21299012788134175955...  2 log 2  6  4 log 2  
(2 k )! 2 k k
k 1

(1) k 1  2k 
 
 
=

k
k 1 8 k
k 
.21271556360953137436... 




.2130254214972640800...

 ((k )  (k  1))
k 2
.2130613194252668472...

.2131391994087528955...

2
1 
e


 

LY 6.114
2
( 1) k 1


k
k 1 2 ( k  1)!


7 (3)
 ( 2k )




2
k
4
k  0 4 ( 2k  1)( 2k  2)

Yue & Williams, Rocky Mtn. J. Math. 24, 4(1993)
.2131713636468552304...

 
1
3
 erfi  
9 / 4  e erfi
8e 
2
2
2

e
 x2
sin x cos2 x dx
0
( k  1)
1
1

hg( )   2
  ( 1)k 1
7
7k
k 1 7k  k
k 1
1
.21324361862292308081... 

7


1
1
1 .21339030483058352767...   k

 2  k 1
1
k 1 2  ½
k 1 2

.2132354226771896621...




3 .21339030483058352767... 
.21339538425779600797... 
 0 (k )

D(k )
,
D
(
n
)

 0 (k )

k 1
k
k 1 2
k 1 2
k 1

4
log 2
1


  2
9
3
k 1 4k  6 k




1
=
  x 1  x 2 log x dx 
GR 4.241.10
0
 /2
=
  log(sin x ) sin x cos2 x dx 
0
sin 4
.2134900423278414108...  
2 
( 1) k 1 4 k
 

1
k  0 k !( k  2 )



2 log
1879
( 1) k 1

 

.21355314682832332797... 
3
7560
k  1 k ( 3  k / 3)

( 1) k 1 2 k k 2
1
.21359992335049570832... 
( 2 cos 2  2 sin 2 )  

( 2 k  1)!
8
k 1
(1) k 1 2k 1 k 2
 
= 
(2k )!
k 1
1 cot 3 
1
.2136245483189690209... 

 2 2
6 2 3
k 1 k   3



.2136285768839217172...

1  11 13  
k

, ,1  
1F1 
11  2 2  k 1 k!(2k  9)


 
.2137995823051770829...


511e 945 
erfi 1 

32
64
1 8
1
 arcsinh

9 27
2 2
( 1) k 1

k 1 k  2 k 
2  
k

GR 4.384.11

.21395716453136275079... 

1


4
k 1 6 k  k


 5k ((5k )  1) 
k 1
1 .2140948960494351644...

.2140972656978841028...

6k
k 1

.2140332924621754206...
(3k  1)

k
k 2
5

5

5


1
 3
1
1 
1  3 


cosh

1/ 3
2/3
2
2
2((1) )((1) ) k  2  k 


dx
1
1
1
   k  x 
e  1 e k 1 e 1 e

1 .214143334168883841559... 
   ( k )  ( k ) ( k )  1 
k 2
.214252808872002637721... 

1  (1)  4 
7
     (1)     2 Root 64  48#1 #12 &,2 
81 
9
9



 
=
x
1

 .2142857142857142857
=
1 .2143665571615205518...

3

14


k 2

.214390956724532164999... 
2
27

.21446148570677665005... 


log k


 3
36
x

k (2k )

9k
k 1



dx 

1
 ( 2 x)  1
x

9k 2

2
 1) 2
 (9k
k 1
dx



( k )  1
 1 
1  log 2 1   3
1 
3 1 
 1 
 
   
  
  
 
2
 2
2
8  2
2
2
2  


( 1)k

 
=  3
k 2 k  2 k


( 1) k
.21460183660255169038...  1 
 

4
k 0 2 k  3

( 2 k  1)!!
 
= 

J387
2
k 1 ( 2 k )! ( 4 k  1)
.2146010386577196351...


 ( x)  1
2

log x dx
 1  x 3
3
 
=

1
2 .21473404859284429825... 
dx

4
x  x2
 /4

0
sin 2 x dx

cos2 x
1
arcsin x
 (1  x)
2
dx 

0
3
14
4
1 .2147753992090018159...

log 2  3 log 3  3   log ( x) dx 
3
GR 6.441.1
.2148028473931230436...

 

3 log 2
 log csc

2
2 2


 (2 k )
1 

=  k
   log 1  2  
 8k 
k 1 8 k
k 1

log  
(1) k
k
k  0 4 ( 2 k  4)

.21485158948632195387... 
8 log 4  8 log 5  2 
(( k )  1)2


k!
k 2
(1) k  1
.215232513719719453473... 
 p , where p is the kth prime
p prime 2  1

.21506864959361451798... 

.2153925084666954317...

 ( k )(( k  2)  1)
k 2
.2154822031355754126...
.2156418779165612598...


4 2
1
1
 

12
3 6 2
24  6 
  7( 3)

 
=
.2157399188112040541...

.2157615543388356956...

1 .2158542037080532573...

2 .21587501645954320319... 
.21590570178361070...

2
 /4
0
0
2
 sin x cos x dx 

sin 3 x
dx
tan 2 x
1
  arcsin 2 x arccos2 x dx
0

4
4k
 

3
4
27
k  2 ( 2 k  1)

( k )  1
( 1) k k ( k  1)

2k
k 2
4

k2
31(5) 
7(3) 



5
4
6
k 1 ( k  1 / 2 )

3
4
H
log   ( 1)k 1 kk 
4
3 k 1
3
12
2


2
( 2)

  (k )  1

 2



k  2   ( k  1)  1
2
.21565337159490150711... 

2
 /4

1   1  e
 e
     
 
 2
2  2 
( 1) k

k 0 k  e


1 .21615887964567006081... 
  ( k ) ( k  1)  1 
k 1
.216166179190846827...
1 .2163163809651677137...
1  e2
sinh 1



4
4e


 (1)
k 1
k 1
2k

(k  2)!


1
1
1
( 1) k 
cos  sin  1  sin    1 

4
4
2
(2 k  1) 
0 



 
=

.21634854253072016398... 

( 1)k
( 1)k




k
k
k  0 ( 4 k )!! 4
k  0 ( 4 k  2 )!! 2  4
4  2G 

13 .2163690571131573733...

k
4
2
4

3
8



1

k 1 3 ( k  1)
3 log 3  3 log 2  1  

=
=
( 1)k

k
k 1 2 k ( k  1)
1
x

0 log1  2 
k2  3


2
k 1 k  2

2 .21643971276034212127... 
(1) k 1 k 2 (k  1)


4k
k 1

(( k )  1)2 
k 2
.2163953243244931459...
7 (3)

2

k

 1  k
k 1
2
GR 1.513.5
1 

 2

2
1
  log(3  2 2 )  

J266
32
k  1 (8k  7)(8k  3)


1
1

1 .2167459561582441825...  log 2 
 2
 
6
k 1 4 k  3 k
k  0 ( k  1)( 4 k  1)
1 log 2 1 
 1 
 1 
.216823416815804965275... 

     i      i       i      i  
 2 
 2 
2
2
8
.2167432468349777594...





 
=
.216872352716516197139... 

 

 

cos2 x
0 e x e x  1) dx 
 (3)  2 log 2 2 log 5  2 log 2 log 2 5 
1
 5

.2169542943774763694...
 4

5
  Li3    Li3 
Hk
k 2
k
k 1

 sin  2
2


  1  2 
k 
 2
2 
=
5

48 .2171406141667015744... =
.21740110027233965471... 
93 (5)
x 4 dx


2
sinh x
0
2  9
4

k

2
k 1 ( k  1)( k  3)


1 .2174856795003257177...

1
k  0 k !( k  4 )!
24 I4 (2)  0 F1 (; 5;1)  24 
log3 5
4
 4
 Li2   log 
2
5
 5
1
.217542160680600158911... 
 cosh2   sinh    2 cosh   1 csch2 
4

( 1)k
.2178566498449504...
  3

k 2 k  4

1 2
(1) k  1 Hk
.218086336152111299701... 

 log 2  

2 24
k 1 k ( k  2 )
( 1) k 1

2
2 
k  1 ( k  1)


i
 2  e 2i   2  e 2i  
4
 2
2 .2181595437576882231...     
 5
1 .21816975871035137...
 j2  2  g2 
.218116796249920922292... 
.2182818284590452354...


 sin k cos k  (k  1)  1
k 1
J311

5 
1
1

e  

2 k 1 (k  2)! k 0 k !( k  2)( k  4)
( 1) k e 2 k 1
k  0 ( 2 k  1)

22
1
 (3)  48 log 2 
 8  64   4

3
3
k 1 4 k  k

k 1 ( k  3)
=  ( 1)
4k
k 1

1 .218282905017277621...
.2185147401709677798...



 
37 .2185600000000000000
1 .2186792518257414537...

arctan e  
116308  Fk 2 k 3

3125 k 1 4 k
2 2


sinh


2 2
=

 1
2 

k 1

 (k )
k
.218709374152227347877...   ( 1)

k (2k  1)
k 2

6
6
H
.2187858681527455514...   kk 
log
5
5
k 1 6
2 .21869835298393806209... 
 arcsin k
2 log 5
136 4 arccsch 2
.218977308611084812924... 


625
625 5
3 .2188758248682007492...

1
  k
k 1


2
1
 1 
arctan
 log1    
k
k
 k 


k4
( 1)

 2k 
k 1
 
k
k
.21917134772771973069... 
cosh 1


erf (1)  erfi (1) 
2
8
( 1)

.219235991877121...

 kk
3

Hk

k 1 k ( 2 k  1)

8 log 2  4 log 2 2  2(3)  

 Ei( 1)     
( 1)k

k 1 k ! k

=
2
LY 6.423
1
log dx
1  ex
1 e x     0 x dx

=
1


.2193839343955202737...
 1  dx
2 4
x
 sinh x
k 1
k 1
1 .21925158248756821...

dx
e
0
ex


 
=
 xe
x  ex
dx
0
2 .2193999651440434251...

.2194513564886152673...

(3)  (6) 

k 1
9 .2195444572928873100...

.21955691692893092525... 
1  1 1
 Lik     
2 
  2 2
k
85 
2  ( 2 )   coth

2
k 1  ( 2 k  2 )
=  ( 1)

4k
k 1
   ( k  2) 
=  Re 

k
 k 1 (2i ) 


 

 


 4k
k 1
4

.21972115565045840685... 


e
5
1
 1  dx
 1  dx

  cosh 2  7   cosh  4
x  x
 x x
4 4e 1
21

.2197492662513467923...
 (1)
k 1
 0 (k )
k!
k 1

1
.2198859521312955567...

1  ci (1)  cos1      x log x cosx dx
0

.219897978495778103...

1

 k2
 ((k
k 1
2
 2)  1)
3 7
2265 .21992083594050635655... 

4

( 1) k 1 H (3) k
.2199376456133308016...  
4k k
k 1
.21995972094619680320... 


 
=
3
 2
 2 log 2


6
24
2

2
2
log x sin x
dx
0
x
.22002529287246675798... 
1
J (1) 
2 1
.220026086659111379573... 
log 2 
 

 2 log 2
24

 log 2 2
2
log 3 2  ( 3)



6
3
(1) k 1 k

2 k
k  1 ( k !) 4

i 
i 
1 

  1 
 
   1 

4 
2 3
2 3 
 3  i 3
1   3  i 3
 
 
 
4  6 
 6  

 
=
(1) k  1

3
k  1 3k  k


 
.2200507449966154983...

.220087400543110224086... 

k 2
.2203297599887572963...

 

 (2 k  1)
2k k
J385

1  1


    k log 1  2   
 2k  2k 
k 1 
 1 
 1
coth   tanh  
2
4
2 2 4
2 2


1
  (2 k ) 
= 
  Re
2
k 
k 1 4 k  4 k  2
 k 1 (2i ) 

coth  

=
 ( 1) k

(4 k )


 4k
1
1
4
k 1
i
 7 1
.22037068922591731767... 

  (2  i )   (2  i )    (1) (2  i )   (1) (2  i ) 
4 64 8
16

1
 
=  3
4 2
)
k  2 k (1  k
k 1

2
 (2 k  1)!! 1
 2 log 2 
 



k  1  ( 2 k )!!  2 k

4G
1 .2204070660909404377...

.2204359059283144034...


 
=

 


 
=
k
4


(3)(4)
1 (1)  1  4 2

   
6
27
 3
1
5 2 (1) 2 / 3


Li2 (1)1 / 3  Li2  (1) 2 / 3 
3
108
2
2/3
(1)

Li2 (1) 2 / 3 
4

1   1
( 1) k
 5 







  
  
 6   k  0 (3k  2) 2
36   3








 
=
.220496639095139468548... 
.22056218167484024814... 
.2205840407496980887...

.220608143827399102241... 
1
x log x
log x dx
1 x3  1   0 x3  1 dx 

 3
8 2  2 
3
e 
  
dx
2 x2/3
0
1

2 .220671308254797711926... 



( 1) k 1 kHk

k !2 k
k 1

2
log 2

1
 x tan  xdx

1


1
 log1  x  log
2
1

  2  x tan  x dx 
0
arctan 2
4 2  4 arctan 2 2  3 log 2 3 
12 2
2
x
dx 
  x log
 2x  3


x dx
0
1
0
2

0
1 
2 

 1   ( k )
k 2

0
x dx
 e x
2x
 Rei   
2
1
3 ( 3)
2
 4 log 2 
 6    log(1  x ) log 2 x dx
2
3
0
cos

1 .2210113690345504761...
e
1 

 1
   2  Ei    log 2  1 
 2

2 e
3 (3)
6 .220608143827399102241...   ( 2)  4 log 2 

2
.2206356001526515934...

1
( 27  3 3  2  2  27 log 3  8(3)) 
16

1
 
=  3
k 1 k ( 3k  2 )
 2 205
.2213229557371153254... 

  (1) (5)  (1, 2, 5)  (2, 5) 
6 144
.2211818066979397521...

1 .2214027581601698339...

e
5

2 .22144146907918312350... 

2

=
 
=
x

2
x dx
 x 4  1 




dx

1
Marsden p. 231
4
 /2

tan x dx 
0
d
 1  sin  

log x  
dx
 x
Marsden p. 259
2
0
2
=
2
0
1
2
cosh x 2 
0 cosh x dx 



=

1 
4  dx 

 log1  2 x
0
.22148156265041945009... 
.22155673136318950341... 
.221565357507408085262... 


 1  1
  1   
   


8




 (k )


k
k 2 



k 1
2
1

k  k
2





1 5
 7  3 5 arctan





3
5
arc
coth
2
2
arc
cot
1
5

4
10 3  5 

k 1

(1) Fk Fk 1

 
= 
4 k (2k  1)
k 1

1
1
log(1  x ) log 3 x
dx
0
x
1 ( 2)
( 2)
.22156908018641904867...   ( 3)  1   ( 2  i )   ( 2  i )  
4

x 2 sin2 x
 
=  x x
dx
e (e  1)
0
6 .221566530860219558...


6 (5) 

log 2 1
( 2 k  4 )! k 2
 
2
8 k  2 ( 2 k )!

( 2 k  1)!!
= 

2
k  2 ( 2 k )! ( k  1)



1
1
  3

  ( ( 3k )  1)   ( 3)  1   6
3
k 2 k  1
k 1
k 2 k  k
.22157359027997265471... 
.221689395109267039...


 
=

 
=

 
=

 
1
1 i 3  5  i 3
1 i 3  5  i 3 
   1 

 


3
2
2
 2  
 2 
 1 1
 3  i 3
 3  i 3
 
  (1  i 3 ) 
  (1  i 3 ) 
3
6
 2  
 2 

3


2
3 3i  3

 5  i 3  1
 5  i 3
2
   
 

 2  3
3 3i  3  2  



1
  1  (1) 2 / 3   2  (1)1/ 3   (1) 2 / 3  2  (1) 2 / 3 
3

1
 (2 k )
.2217458866641557648... 

(log 3  log 4  2  )  
6
4k
k 1

2
k
2
.22181330818986560703...  2 log 2  log 2 

 k
6 k 0 2 ( k  1)2
=



1 .2218797259653540443...


k
k 1 
2
k 1
log
k 2

97 6
2
 2 (3)  
22680

1
  4

k  2 k  11
1 .221879945319880138519... 
.221976971159108564...

.22200030493349457641... 

 (2 k  1)  1
k!
k 1
.22222222222222222222
=
2

9

=


 ( 1) k 1
k 1

Hk Hk
k4
k 1

 e1/ k
1
  
  
k
k 2  k
2

1
k
 x 2 arccos xdx 
2 k 0
 (k )3k
9k  1
Bk
.222495365887751723582...  
k 1 ( k  1)!
k 1



1
2
2
1
H
log 
arctan
  ( 1) k 1 2 kk 1 
2
6
3
3
2
k 1
3
2

csc 1
cos 3  12 sin 1  9   k  (22k k ) 
.222640410967813249186... 

32
k 1
2

3e 1
k 1
2 .2226510919300050873... 



4 2e k  0 (2 k )!
.2225623991043403355...

.2227305519627176711...



21 (3)  4


4
16

 
=
 (1)
k 3
.22281841361727370564... 
12 k
 (2 k  1)
k 1
4

k ( k  1)( k  2)( k )
2k
G  log 2 
 

1 
2 
 
 8  4  1 
   1 
4  
2 2

 


8
 ( 2 k  1)
 
= 

 
3
2 k 1
k 1 8 k  k
k 1


2 k  ( k  1)
e2/ k  1
 
8 .22290568860880978...
 

k!
k
k 1
k 1


 (k )
1
3
2 .2230562526551668354... 
(9 log 3   3 )   2
  k 2 
2
k  1 3k  k
k 2 3
2 .22289441688521111339... 

k 1



 
=
1
 ( k  1)( k 
)

(1) k 1
1

1  1
log 5  log 4   ,1,0   Li1    k  k

5

 5  k 1 5 k k 1 4 k
k 0
.2231435513142097558...

2
3


 
1
1
= 2 

2 k  1  2arctanh
9
k  0 ( 2 k  1) 9

.2231443845875105813...


dx

x
1
 4e
0
K148
H ( 3) k
 3 ( k  1) 
k
k 1
.2232442754839327307...


( 1) k 1 k
( 1) k 1


k 1 ( 2 k )!( k  1)
k 1 ( 2 k  1)!( 2 k  2 )

2 sin 1  cos 1  2  
1

 
=
x
2
sin xdx 
0

log 2 x sin log x
 1  dx
dx   sin  4 
2
1
 x x
x
1
e

 
=

.22334260293720331337... 

k 1

.2234226145863756302...

(2 k )  (2 k  1)
(2 k )!

( 1)k
 k ! 1 
k 0

.22360872607835041217... 

(( k )  1)2

k
k 2

267  ( 1)k k 9

e
k!
k 0
1 
1


 2
4
2 e  e

( 1) k 1
=  2

k 1 k  2
98 .22381079275099866005... 
.223867833455760676...

 
.2238907791412356681...

1
csch  2 

4
2
J125

( 1)k
( 1)k 1 k 2
 J0 (2 )  0 F1 (;1; 1)  


2
( k !)2
k  0 ( k !)
k 1
2


 2k 
1  2k 
k
k 1 k
=  ( 1)
 
    ( 1)
(2 k )!  k 
(2 k )!  k 
k 0
k 0
1 .223917650911311696437... 

e  (3)1

.22393085952708464047... 
4


3
2 2
)
k  2 k (1  k
k 3 ( k )
  ( 1)

k!
k 2

3 .2241045595332964672...

2
k

 (2k  1)  1
k 1
k

log(1  k 2 )

k
k 2


 1 3k  1  k 2 
 


e1/ k k 3 
k 2  k

LY 6.115
.224171427529236102395... 
 3  log 8 

 (2 k )  1
 k (2k  1)
k 1

.224397225169609588214... 
 ( 1)
k
 (2 k  1)
k
k 2


.2244181877441891147...

.2244806244272477796...

 
=
.22451725198323206267... 

1
1 
2  


  k  kLi   k

2
2
k 1

kH
7
7
(1  log )   k k 
36
6
k 1 7
5  log 2
 
 2 (1) 
3 3
2


 ( k  1)  1
1
1
 1


  k  k 2 log 1   




k 3
3k 2
k 
k 1
k 2 
 e


 .2247448713915890491...
1 .2247448713915890491...

3
1 
2

3

2
(2 k  1)!!
k 
k  1 ( 2 k )! 3


1  2k 

k 
k

 12
k 0

( 1) k 
1 


6k  3
k 0 

=
.22475370091249333740... 
1 .2247643131279778005...
.2247951016770520918...

 1  dx
 sinh x  x
5

1
1 cot 3 
1

 2 2
18
6
k 1 k   9

1


1


cot
  2

2 4 2
2 2
k 1 8 k  1




.22482218262595674557... 
.224829516649410894...
8
e 
e



2 3
cot

3


k 2
1
3k  1
2



k 1


k 1
(2 k )
8k
( 2 k )  1
3k



1
1
(32  3 2  24 log 2  27(3))   3
81
k 1 k (2 k  3)
.225000000000000000
=
.22507207156031203903... 
.225079079039276517...
.225141424845734406...
9
40


log 2



4 2 3
2
1
 arctan x dx
3
0
 3
1
  1    1   
 2
 4
 4
1
1 

  dk
 2G    log 2  1     E ( k )  

2 0
2 k
2
1

GR 6.149.1

.2251915709941994337...

.2253856693424239285...


2k

(k  1)!!
k 1
3 (3)
1
  Li3  1   Li3 (i )  Li3 (i ) 
16
4
 (1)1


 
=
1
log 2 x dx
x log 2 x dx


1 x3  x
0 x 2  1


 
x 2 dx
=  2 x
e 1
0

2e
1


2e  1 k  0 ( 2e) k
 3
1  1 
1 .22541670246517764513...      2    
 4
 4


 (k )
 (k )
.22542240623577277232...  

 
k 1 k ( k  1)
k 1 k  1
1 .225399673560564079...


1 .22548919991903600533... 
2  1
 coth(log  )
2  1
1
.225527196397265246307... 
4  2 arccot 2  2 log 3  2 log 2 
18


  (1)
k 1
k 1
kH2 k

2k
1
8
4  2
H2 k

  4 log  2 log
k
7
7
4  2
k 1 8

.2256037836084357503...

1 .2257047051284974095...



 (3k ) 
k1
.2257913526447274324...

.226032415032057488141... 
log


2

1


8 6

 ( 2k )
2
k 1

2 k 1
k


arctan x
dx
x5
1

 (1)
k 2
k
log k 
K ex. 207

3 .22606699662600489292... 
 (1)
k 1
(3 (2 k )  1)
k 1

8 .22631388275361511237... 
.2265234857049204362...
2 .2265625000000000000

erfi 2 

4k
 k !(2 k  1) 
4
k 0
e

12
285
1

 1
=
  ,4,0  Li  4   
5

 5
128


1


.226600619263869269496... 
.226724920529277231...
.22679965131342898...
.226801406646162522...
 
.2268159262423682842...
1 .2269368084163...


k4

k 
k 1 5
.226589292402242872682... 


1
 3  5 Li   3  5   1  5 Li  1   2 Li  
2
2
2


 1 5
8
51 5 
4



k 1

(1) Fk Fk 1

 
= 
4k k 2
k 1




1  5
  
4  4


2

 



x e
4 x4
dx 
0

1

 36k  5
8 3
k 1

k 3H k
287 153
77



log 2  log 2 2   k
4
2
2
k 1 2 ( k  1)( k  2 )( k  3)

 36k
2
7
2 2
 4 log 2  2 log

2
8
2 2

1
=  2

k 1 8k  k

1
H

72 log 2 2  168 log 2  149   k k

9
k  1 2 ( k  4)
 root of ( x )  5



cot
.22728072574031781053... 
2 
2
  2, 3,  
27 
3
.227350210732224060137... 


 (k  1)  1
k 1
3k k
 .22740742820168557...

Ai(-2)
.22741127776021876...

3  4 log 2  
=

log 2 x
1 x 3  2 dx 

1 
1 
   log1  
 3k 
k 1 k

1
( 1) k 1



k
2
2
k  0 2 ( k  2 )( k  3)
k 1 k ( k  1)

( 2 k  1)!!

2
k 1 ( 2 k )! ( k  1)


1 .22741127776021876...

4  4 log 2 
4
k 0


 


=
1


k 1 k ( k  ½ )
( k  12 )!
1 )!
2
k 1
3
 2arcsinh

2
k
 2k 
1

2 
( k  1)  k 
(1) k 1


2
k 1 k k  1 / 4


  (1)

k 2
k
 (k )
2k  2


 
1 .2274299244886647836...
4 .2274535333762165408...
1 .2274801249330395378...
=
 k (k 
( 1) k 1 3k

 2k 
k 1
k2 
k

(1 / 4)
1
    
  
 3 log 2  
(1 / 4)
2
 4


1
1
Hk  1
2
4
 3 (5)   6 
  (3) 
 12 log 2 2  
 2  3  4

6
72
k
k 

k 1 2 k  1 k

1 .227558761008398655271... 

 (k ) (k )  1
k 1
k 2

1 .22774325454718653743... 
Hk
 k !k
k 1


1 .2278740270406456302...


2


1
 k !(k  1) 
k 1
 1

log(2  2 )  
,1, 0 

 2


1
3k  1

k  (k )
.228062398169704735632...   ( 1)
   k arctan


2k  1
3k 
k
k 2
k 1 
1 .227947177299515679...

2 .2281692032865347008...

16 .2284953398496993218...
.2285714285714
7





 e  1 
erf 2  
2


2
=
8

35

p prime

2k

k  0 k !!
1  p 
2 2
1  p 2  p 4
 

  2 (k ) 
2 ( k )
2 ( k )





1




  k s
k  2   ( 2k )
s2 k 2
k  2 k ( k  1)


2
11 (5)
1
.22881039760335375977... 
 MHS (3,2)   3 2  MHS (2,2,1)
 (3) 
2
2
k  j 1 k j

2 .22869694419946472103... 

3 log 2 2  7 2
log(1  x 2 )
 
  4
dx 
x  x3
4
2
48
1

G
( 1) k
.2289913985443047538... 
 

2
4
k 0 ( 4 k  2)



1
1
 
=  
2 
2 
(8k  2) 
k  1  (8k  6)
.22886074213595258118... 



 
x dx
=  2 x

e  e 2 x
0


 
=
x log x
 1 x
4

log x dx

3
 x 1
x
1
dx 
1
e
.2290126440163087258... 
(log (e  1)  1) 
e 1

H (2) k
.22931201355922031619...   k
k 1 5 k
( 1) k 1 Hk


ek
k 1


.22933275684053178149... 
.229337572693540946...

.2293956465190967515...

.22953715453852182998... 

 
=
.22955420488734733958... 
.229637154538521829976... 
.22968134246639210945... 
.2298488470659301413...


2 .2299046338314288180...


=

7 (3)  2
log 2 x
dx

 
2
x
x
8
12
(

1
)
(

1
)
1

1  1
1
 log 1     k
2  e  k  1 2e k

ek
I0 (2 e )  0 F1 (;1; e)  
2
k  0 ( k !)

3
k 1
   log 2   ( 1) k
 ( k )  1 
2
k
k 2

1
1 
 

 log 1   



k
k  1
k 2 

1
1 1
1
( 1) k 1 k
cos  sin  

k
8
2 4
2
k 1 ( 2 k  1)! 4

3
k 1
 (k )  1 
   log 2   (1) k
2
k
k 2

( 1) k 1 k
1
1
sin
 
k
2 2
2
k  1 ( 2 k )! 2

1 1  cos1
( 1) k 1

sin 2 

2
2
k 1 2 ( 2 k )!

 1  dx
1 sin x 2  x 3


( 2 k )
1
1

cosh


2
k
k 1 ( 2 k  2 )!
k 1 k
GR 1.412.1
.2299524263050534905...
.2300377961276525287...
.2300810595330690538...
.23017214130974243...




.230202847471530986301... 

24 3 log 2 



25
5
10

2 1

2
4

0
k 1
xarccos x dx

(1  x 2 ) 2
1
 5
 cos

4
2


k 1
26

e
GR 4.512.8
5
 1  (2k  1)

1

 1 
erfi 
 
1/ 8
2 2
2e
2
16 
1
 k (4k  5) 
2

 

( 1) k 1

 2k 
k 1
k !  2 k
k

( 1) k


k
k  0 k ! 2 ( k  3)

log10

10
1
  ( 1)k
3 .23027781623234310862... 
 2  csch   2
2
2 k 0 4 k  1

1
.23027799638651103535...   k
k 1 4 ( 2 k )
  
1
.2302585092994045684...

.23030742859234663119... 
   2 3 
k
s  2 k  0 j 1
.23032766854168419192... 
.23032943298089031951... 
.230389497291991018199... 
 
=
j s

5 5
1  2k 
  ( 1) k 1
 
12
k 2 k 
k 1
1
6
1    1  i
 i 
  1  i
 i
 
          
 
 2 
 2
 2
4  2 


1

cos x
dx 
x
 1) 
 e e
x
0
2
1  (1)  1 
 3 
     (1)    
.230636794909050220202...  G 2 

8
32 
8
 8 
.2306420746215602059...

 
log 2

2


 .230769230769230769
=


k4


k
k 1 k
1
  1  2 x
0
3
sin x dx


13 0 e x

9 .2309553591249981798...
5

2
 dx
 cos x 

 x
 /4

0
x dx
sin x  cos x
.2310490601866484365...

log 2

3

=

1
1 .23106491129307991381... 

 
.231085022619546563...

( 1) k 1


k 1  2 k  k
 2 k
k
( 1) k


k  0 3k  3



dx
dx


4
x  x 0 (3x  1)(3x  2)

e
0
J292
dx
3
2x



  

  
cosh cosh   sinh    sinh cosh   sinh   
 2
 2 
 2
 2 


 / 2
ei  ee 

( k )
 

2
k 2 k ! k
=
1 .231141930209041686814... 
i
6
1890
 (3) 2


2

H ( 3) k

k3
k 1

.231274970101998413257... 
 log (2 k  1) 
k 1

.2312995750804091368...

k !!
 (k  3)! 
k 0

1 .23145031894093929237... 
2
12
 2 log 2   (1  log 2 ) 
( k  1) Hk

2k k
k 1



1
(1) k 1 k


 e1 / 
k! k
k 1
1
.2316936064808334898...  Ai  
 2


k 1
1
 
 1
.23172006924572925...
  2  ( k )  1     log 1    Li2   

 k 
k
k 2 k
k 2 
.23153140126682770631... 

2 .23180901896315164115... 


1
  2   ( k )  1
k 2
.2318630313168248976...

2 log 2  2  


k 0
1 .2318630313168248976...

2 log 2  2   1 
 (k  1)
2k



 ( k )k
2k
k  ( k  1)
2 .2318630313168248976...  2  2 log 2  2   

2k
k 0

k  1 1 ( k )  ( k )
1 .23192438217929238150...   ( 1)
2k  1
k 1
k 1


.2319647590645935870...

.23225904734361889414... 
.2322682280657035591...
.2323905146006299...

1
2
 K (1) 
6
9
x
2
arcsin( x 2 ) dx 
0

351  14  3  126 log 3
1


588
k 1 k (3k  7)

k log k
  ( 1)
k ( k  1)
k 2

.232609077617500793647... 
1 .2329104505535085664...

9e
k
 12  

2
k  0 k!( 2 k  8)

6 (3)

3


 e
0
dx
x1 / 3
1


x 2 dx
0 e x e x / 2  1 
16 (3)  18 


1
3 .2329104505535085664...
log 2 x dx
 16( ( 3)  1)  

x
0 1
=
.23300810595330690538... 
4 .23309653956994697203... 
.2331107569002249049...

.233207814761447350348... 
16 (3)  16 
1
 5
 cos

4
2
2

(3) 





5
 1  (2k  1)
k 1

2



3
32
1
log 2


   arctan x log x dx
4 48
2
0

(2 k )
4k

1 .2333471496549337876...

1   3 coth  csc h 2   Im ( 2 ) (i )
1 .2334031175112170571...

arcsinh 
2



1 .233527691640341023900... 
I
k 0
2
.2337005501361698274...

x 2 dx
0 e x / 2 e x / 2  1
2
k 1


2 1   3 
1
    i (1 / 2 )   
2
 2
 2

.23333714989369979863... 
GR 4.512.8


8
k
(1) J k (1)
1 

1
 (2k  1)
k 2
2

(k  1)( (k )  1)

2k
k 2


GR 4.593.1
( k  1)!( k  1)! 2 k
= 

( 2 k )!
k 1


 
1

 
x 2 log x
=  
dx 
2
0 1 x

 
=

log x
dx
4
 x2
x
1
1 .2337005501361698274...

2
8

k 2 ( k 2  1) 
k!


2
3
k  2 ( k  1)
k  0 ( 2 k  1)!!( k  1)

=
Li2 1  i   Li2 1  i 
=
 (2 ) 
=
J276

3(2) 
1
k 2 ((2 k )  1)



2

k
(
)
4
2
1
k 0
k 1

1
1




2 
2
(4 k  3) 
k  1  ( 4 k  1)

=
2k

k 1  2 k  2
 k
k

=


 (k  j )
2k  j
j 1 k 1
1
=
log x dx
x2  1
0

GR 4.231.13

=
log x dx
x4  1
0


=
arctan x dx
1  x2
0


=
arctan x 2 dx
0 1  x 2

=

arcsin x dx
0
1  x2
1
=
GR 4.538.1
dk
 K(k ) k  1
GR 6.144
0
1
=
 E ( k ) dk
0

.2337719376128496402...

 (k )
 k !2
k 2
k
GR 6.148.2


 (k )


 (2 k )!!   e
k 1

k 1
1/ 2 k
1
1

2k 
 2   3 5  1    1  dx
1
PFQ  ,  , ,     sin  3  2 
4
 x x
 3   2 3  4 
1
1
1 
1 
1 



 
   1 
.23405882321255763058... 
cot
 
   1 

2
2 2
2 2 
2
2 


k 1
 ( 2 k )   ( 2 k  1)
 
= 


3
2k
k 1 2 k  k
k 1


1
 (k )
.2341491301348092065...   k
 0k 
6
k 1 6  1
k 1
.23384524559381660957... 


.234163311975561677586... 
2
9
 (1)  1 

 2
     (1)     3 log 3 
 3
 3



.23416739426215481494... 

 ( 3k )


 36k
6

Hk
  2k 
k 1

 (2k  1)
k 
1

 6k 3
6
k 1
( 1)k
.23437505960464477539...   k ! 
k 0 2

 ( 2k )  1  
3
1 

   k arctanh  1
 log 2
 
.23448787651535460351... 
k 
2
2k  1
k 1
k 2 

log k
.23460816051643033475...   3
k 2 k  k
1 
1 
 (3) 1  

.23474219154782710282...    
   1 
 
   1 

6
2 
6
6 
k 1

k

1
 ( 2 k  1)



5
3
6k
k 1 36 k  6 k
k 1

( 1)k 1
 1  J0 (1)  

2 k
k 1 ( k !) 4
1
4
 3 


 16    4,   
1 4


16
2
k 1 ( k  2 )


 
.2348023134420334486...
.2348485056670728727...
=

.234895471785399754782... 
1  ( 2)  4 

 
7

     ( 2 )    16 3  cos  2 3 sin   
729 
18
18  
9
9



 
=
log 2 x dx
1 x 3  1  x 3
2
.23500181462286777683... 
.23500807155578588862... 
3 log 2 log 5
dx

  3
2
3
x x
1

2 sin 1
sin k
  ( 1) k 1 k 
5  4 cos1
2
k 1
.235028714066989...

 
.2352941176470588

 
1 .2354882677465134772...
1 2
 3
 3
  12   3  2 sin 3  csc
sec

36
2
2

( 1)k
=  4

2
k  2 k  3k

4
sin 4 x dx
=


17 0
ex


=


1

2 cosh(log 4)

 (1) e
k
(2 k 1)2 log 2

J943
k 0
  3
 
3 sech
 2 

k3
 (3k )  1


exp


3
k
k 2 k  1
k 1



 

 

 
k3
= exp  log 3

k 1
k 2

z (k )
=  3
k 1 k
=
.23549337339183675461... 
 5 i 3  5 i 3
 

3  ( 1)1/ 3   ( 1) 2 / 3    
 2   2 
3 (3)
 2 log 3 2 
4
1
log 3 (1  x )
0 x 2 dx 

( 1) k 1
1




k 1
k
k
1
k 1 2 ( 2  1)
k 1 2

2 1  ( 1) k 1
( 1) k k  ( 1) k 1

 


e 2 k 1 k !( k  2 ) k  2 ( k  1)! k 1 ( k  1)! k !
89

CFG D1
40
1
 1
 1 
Li2      4 log 2 2  4 log 2 log 5  log 2 5  Li2    
2
 4
 5 
k 1


Hk
( 1)
 k 2 

k
k 1 5 k
k 1 4 k

.23550021965155579081... 
.2357588823428846432...

.23584952830141509528... 
.235900297686263453821... 

 
=

 
=
1

0
log x
dx
x4

.2359352947271664591...

5 .23598775598298873077... 
.23605508940591339349... 
k
k
k 1 ( k  1)! e
(1  e ) e1/ e  e  
5

3

dx
 1 x
6/5

0


1  log 2( 2  1) 
2  log 2
 
2 2
 /4
 sin xlog(sin x)dx
0
.236067977499789696...
52 

1
3

( 1) k 1  2 k 

5 1  
 
k 1 2 k  1  k 

 ( 3k  1)
= 
k 
k  0  ( 2 k  2 ) k !8

1 .236067977499789696...

 
Berndt Ch. 3, Eq. 15.6
1  2k 

k 
k 0 5  k 

2 .236067977499789696...

4 .236067977499789696...

 3  2  1

12 log 2  9 log 3  3 3 
.2361484197761438437...


5  2  1    2  
3
2
6

 224
(2)  5 
 7 (3) 
2 .236204051641727403...
      2 

 2   27

1

2
1
1 .2363225572368495083... 

cot 
 2

4 2 6
3 k 1 3k  2


 6k
k 1
3
1

 k2

log 2 x
1 x3  2 x 1 

( 1) k 1
.23645341864792755189...  3  2 cos 1  2 sin 1  

k 1 ( 2 k )!( k  1)


3 e
k2
k2
1 .2365409530250961101... 


k
4
k 0 k ! 2
k  0 (2 k )!!
.236352644292107892846... 
1 
   2, 3,
32 
csc 3
1
1 .236583454845086541925... 


6
18

.236585624515848405511... 
2G 
2
12
1
 
2
( 1) k 1


2 2
k 0 k   9

  log 2 
log 2 2

2
(1) k 1 H k


k 1 k ( 2 k  1)



 .2368775568501150593...

 
5 3
13 (3)
1  ( 2)  5 
 1 
     ( 2 )    


18
216 
81 3
6
 3 


(1) k
x 2 xdx
= 2

3
0 e2 x  e x 
k  0 (3k  2)



 
.2369260389502474311...
=
log 2 x
1 x 3  1 dx 

25
( 1)k 1 k 6



64 e k 0 k ! 2 k

2 .2369962865801349333...

4k
  2k 
k 1
.23722693183674748744... 
 k 4
k 
1

9 (3)   2
log 3 x
x log 3 x


 
4
( x  1) 3
( x  1) 3
1
0
log(3  3 ) 
 4  ( 1)k
.237462993461563285898... 
 

2 3 k 0 k  5 / 2
.2374007861516191461...

 /2


=

0
.23755990127916081475... 
=
2
24
sin 2 x
dx
(1  sin x) 2



3 2  5  1
  2 5  2 
log 

4
2


 1
1
 3 3 1
PFQ  ,1,1,  , ,   
4
 2   2 2  4 
Berndt Ch. 9
(1) k 1


k 1  2k 
2
  (4k  2k )
k 

16 3

3

k3
17 .2376296213703045782...  

k 1 ( k  1)!!

1
 2 137
.23765348003631195396... 


 2
30 1500 k 1 k ( k  5)
9 .2376043070340122321...

.2376943865253026097...


CFG D11

  (( jk  3)  1) 
j 1 k 1


98 .23785717469155101614... 
 .2378794283541037605...
.2379275450005479544...

2
24  log 2 2   
12 



0
e x x 3 dx
(e x  1) 3
1  (1)  1 
(1)  2  

        
 6
 3 
144 



GR 3.455.2
log x dx
3
 x 3
x
1
 ( 2 )   ( 3)   ( 2)  2  

 ( k  1)
 k ( k  1)
k 1

.23799610019862130199... 
.2380351360576801492...
.238270508910354881...
6 .238324625039507785...
1


3
k  2 k log k

  (s)  1ds 
3

( 1) k
e
 1 
erf 

 
 2  k  0 k !!
2

1 146
1
 8 2arctanh

 k

2 15
k  0 2 ( 2 k  7)
 9 log 2 

e
2

.2384058440442351119...

9 .23851916587804425989... =
2
e 1
 2 (2 2  3 log 2 2)  log3 x dx


2x2  1
16 2
0
2

1 .2386215874332069455...

 (1)k 1
 ( 2k )
k 1

.2386897959988965136...


.23873241463784300365... 
.238859029709734187498... 
.2388700094983357625...

2 .2389074401461054137...

2 .23898465830296421173... 
.2391336269283829281...

 
.2392912109915616173...

=

k!



  1  e 1 / k 
k 1
2
( 1)k


2
k 0 k  k  1
3

4

 



(1) k 1 k 2 (2k )


 2 coth     coth  3 csc h 2   

32 
2
2
2
4k


k 1

k3
1
3

(e  )  
e
16
k 1 ( 2 k  1)!

k H (3) k


2k
k 1
(3)  (5) 


( 1)k
(2 k  2)

2 cos1  sin 1  
  ( 1)k
(2 k )!
k  0 ( 2 k )!( 2 k  3)
k 0
e
log 2 x coslog x
dx 
1
x
Ei( 1)  log( e  1)  1   1/ ke k 
( k 3 )

k3
k 2

1


3
k 2 k  1

.23930966947430038807... 
=

1


6
3
k 2 k  k

=
  (3k )  1
.2394143885900714409...
1 cot 2



8
4

2 .2394673388969121878...

.239560747340741949878... 
 
=
k 2
12
1
 k3

  (9 k  3)  1
k 1

 k
k 1
2
1
2
4


  (9 k  3)  1
k 1

2(3)  2( 4)

 (3  i 3 )
 (3  i 3 )
1
 2   cot




cot
2
log
2

6 
4
4

(1) k k 2


3
k 1 k  1


k

k 1


.2396049490072432625...

log
e

2( e  1)
J157
2 3
1
arcsinh

.23965400369905365247...  1 
3
2
( 1) k 1 2 k


k 1  2 k 
  (2 k  1)
k


.2396880802440039427...

.2397127693021015001...

1 .239719538146227411965... 
.2397469173053871842...

.2397554029243698487...

.23976404107550535142... 
1

8
k 1

1
1
( 1)k 1
sin   k
2
2 k 1 4 (2 k  1)!

1


k 1  k ( k )

log 2 k
(3)   3 
k
k 1

1
( 1)k 1

2 sin 2

2 2 k 1 (2 k )! 2 k
1
1
J2 (2 2 )  0 F1 (; 3; 2) 
2
2
k
3

( 1)k 2 k


k  0 k !( k  2 )!
1
.2398117420005647259...

  ci (1)    log x sin x dx

GR 4.381.1
0
( 1) k

k 1 ( 2 k !) 2 k

=
7
7
3 (3)
 log   12 (2)   log  

4
4
2

 ( 2k )  1
= 

k 2
k 1

1
1
 4 
2
=    k log 1  2   k  

k 
2
k 2 
.239888629449880576494... 

 
1 .23994279716959971181... 

.2399635244956309553...

.24000000000000000000
=
.24022650695910071233... 
1  2 5 log 2



3 30
6
1
log 2 x
0 ( x  1)6 dx 
 1 1
 2 4

k
6
1

  ,  1, 0    k 
25
6

k 1 6

H
1 2
log 2   (1) k 1 k 
2
k 1
k 1
 ,  

2
k 1
Hk

(k  1)
k 1

(1) k 1


 
=
  2k 
k 1
=
  2 k k 2
k 
1
log(1  x)
0 1  x dx 
1
=
GR 4.791.6

2
log(1  x)
log x
log x
0 1  x dx  1 x dx   0 e x  1
1
.2402290139165550493...
ex  1
 2 log(1  e)  2 log 2  1   x
dx
e 1
0
1 .24025106721199280702... 

3
25
(3)
.2404113806319188571... 

5

.2404882110038654635...


1 .2404900146990321114...
1  ( 2)  1 
 3 
     ( 2 )    
100 
 4
 4 
H ( 3) k

( 1)

3k
k 1
 1   1 
 
 

 1 / 3  2 / 3

k 1

 
4
1
1
log 24  8 5
arcsinh  log 2 

2
4
2

k
.2408202605290378657...  1  ( 2 )  2(3)  

3
k 1 ( k  2 )
.24060591252980172375... 


 
=
 ( 1)
k 1
k 1

2 .240903291691171216051... 

k 0
 2k 
(1) k
 
2 k 1
(2k  1)  k 
k 2 (( k  1)  1) 
Berndt 5.8.5
1
 ( k  1)! (2 k )
Titchmarsh 14.32.1
k 1
.241018750885055611796... 
.241029960180470772184... 
2 log 3  3 1

 
3
6
2


 (k  1)  1
k 1
3k

2
3 (3)

 log 2 

2 24
16
1
 arctan x log
2
x dx 
0
1
.241172868732950643445... 
2  2 2  log 2  2 arcsinh 1    arcsinh x log x dx
.24120041818608921979... 
1 2

6 3
0


13 .2412927053104041794...
257 e


32
k 1
( 1) ( k  2 )!

( k  1)! 2 k
k 1


k5

k
k 1 k ! 2
1
.241325348385993028026... 


12
log 3 2  (3)



6
8
2

 (k )
2
k 1
k
k2


1


 log 2 2

5 log 2 5
k

  k 2
4
8 k 2 2 ( k  1)
1
.24145300700522385466... 

 1
1
4
(1  x ) 2 dx
 .241564475270490445...
 log   


1  x 2 log x
0
.2414339756999316368...

 2

=
x log x  x  1
log(1  x )dx 
x log 2 x

0

.24160256000655360000... 

k 1
1 .24200357980307846214... 
1 .24203498624037301976... 
GR 4.267.2
(2 k )
k
5 1
GR 4.314.3

1
2k
k 1 ( 5 )

15  33

6
 2 (1  e)

e(1   2 )
Associated with a sailing curve. Mel1 p.153

e
 x /
sin x dx 
0


.24203742082351294482... 
 (2  (k )) 
k 2

1 .2420620948124149458...

 

=
.242080571455143661373... 
.24209589858259884178... 
1

k
 1) k
k 1

 1 (k )

k
k 1 2 k
 (2

 log 2
k 1
sin8 x
0 x5 dx
2
  ( 2)
  1 
6

 
0
112  9  24 log 2

27

.242156981956213712584... 
 ( 1)
k 1
.2422365365648423451...

3
128


(k )
4k  1
k 1
k 1
 1 ( k )
2k


2
.24214918728729944461... 



327 / 8
log

32
6

2k
log(1  2  k ) 



k
1
k `
x log x
dx
ex  1
1

x log(1  x ) log x dx 
0


x 2 dx
log 2 x dx

0 e2 x  e  2 x 1 x 3  x 1 
( 1) k

3
2
k 0
1
3
 c2  (3  (2)  (3)   ) 
6

.24230006367431704...

.242345452227360949...

k
Patterson Ex. A4.2
( 1) k 1 H e k

k!
k 1

1
3 .24260941092524821060...  

k
k  2 (log k )

.24241455349784382316... 
 2k 
k
 
k  1 8 ( k  1)  k 

.24264068711928514641... 
3 2 4  
k

18  3 2
4 .24264068711928514641... 
 k  1 k  1


k 
k 
k 1 

8 k 4  3k 3  k 2  3k  1
2
2 .24286466001463290864...  8    13 ( 3)  

k 3 ( k  1) 3
k 1


.24270570106525549741... 
  Li 
k


 
=
 (1) k ((k )  (k  2))
k
3
k 1

.24288389527403994851... 
.243003037439321231363... 
2 (3)  (3)


3

2
(k  12 )!(k  12 )!

(2k  1)!
k 1


1
1  x2
3
0 x 2 arctan( x )dx 
1126
2 log 2
.24314542372036488714... 


2
9  7 5
9
 4 7 2
1
2 .2432472337755517756... 



120
48 128
=

=
 k 3 ((2 k )  1) 
k 1
.2432798195308605298...


 ( k )   ( k  1)
5 e 8 
8 .2436903475949711355...

9G 
.2437208648653150558...

8 log
.2437477471996805242...

k 1
k 2 ( k 4  k 2  1)


( k 2  1) 4
k 2



k!
k 2
.24360635350064073424... 
1
 k (2 k  9) 

( k  1)Bk
e2



2
(e  1) k 0
k!

.24346227069171501162... 


 1/ k e1/ k

1
e 
 2  1

k
k

k 1 

k
 k ! 2 ( k  2)
k 1
k
3
 1

 3    ,1, 3 
2
 2

log(1  2 )


4 2
2 2
(1) k


k
k  0 2 ( k  3)

(1) k


k  0 4k  3


dx
1 x 4  1 
1
x 2 dx
0 x 4  1 


 
=
 3e
0
.243831369344327582636... 
x
dx

 e x
7 1
3
3
 
cot 

2 2
2
2

.24393229000971240854... 
 k ( (3k )  1) 
k 1

.2440377410938141863...

5 .24411510858423962093... 
.2441332351736490543...



k 2

k
 3
   (2 k )  1 

 
k 1 2
1
k 3 1  k  3 
2

 (1   )
1
1
 2   , related to the lemniscate constant
2  4 

(1) k 1
1
1  csch 2





 2 2
4 e 2  e  2
4
2
k 1 k  4



 
=
sin 2 x dx
ex  1
0


log 9 2 log 3


9
9

  k  12 
.24433865716447323985...  
(2 k ) 2
k 1
.24413606414846882031... 
.24470170753009738037... 
J137
2    2 log 2 

( 2 k  1)!!
 (2 k )! k (2 k  1) 
k 1

.244795427738853473585... 
.24483487621925456777... 

.2449186624037091293...

.24497866312686415417... 
6 .24499799839839820585... 
.245010117125076991397... 
.2450881595266180682...

 
G
8

 2 log 2
32

35 (3)
 
128
 /4
 x log sin x dx
0

1
1

4 sin 2   k
4 k 1 4 ( 2 k  1)k
e 1

e 1

1
( 1)k

arctan   2k 1
4 k 0 4 (2 k  1)
39 

(1) k 1
 (3) log 3 2

 
4
6
k 1  2k  k 3
  2 k
k 
 3
3 log 3
1
 2 log 2  hg   
2
2
6

k 1 ( k  1)
=  ( 1)

6k
k 1

6

J148

 6k
k 1
1

k
2
1

 
  log(1  x 6 ) dx 
=
0

 .24528120346676643...


2

16
 /4

log 2  G 
4

0
x 2 dx

cos2 x
GR 3.857.3
1
=
 arctan
2
x dx
0
.2454369260617025968...


5

64
1
x
3
arcsin x dx 
0

H H
1 2
7 (5)
  12  2  12 (3) 
  k 3 k 1
72
2
k
k 1

sin 1 1
sin k
 sin 1 log 2  2 cos1   2
.24572594114998849102... 
4
2
k 2 k  1
5
1 .245730939615517326...

3

1
3
1 cosh 2
4k
e2
2 .245762562305203038...



 
 sinh 2  

2 4e 2
4
2
2
k  0 ( 2k )!( k  1)

2 .245667095372459229031... 


1
.2458370070002374305...

1 sin 1  cos1
sin x dx



2
2e
ex
0
e


=
sin log x
dx
x2
1


.24585657984734640615... 
e
x
1
.2458855407471566264...

dx

(1  log x )
 (2)  2  16  12 log 2 


 
=
 (1)
k 1 ( k  2 )
4k
k 1

2 .2459952948352307...

root of  (1) ( x ) 



.2460937497671693563...
1 .24612203201303871066... 

8 .2462112512353210996...

.2462727887607290644...


k 1
2
(2 k )
22

 4k
k 1
3
1

 k2

1

2
k
1
 3
 2 (log 3  2 log 2) log 2  Li2    Li2   
6
 4
 4
68  2 17 

 1

   , 4, 0  
 4

(1) k 1


k 4
k 1 4 k


.24644094118152729128... 
k
k 1

 
=
=


 
3
2 log(1  2 )  2 arcsinh 1  2 arctanh
1 .24645048028046102679... 

1

 k  k2  k 1
4
1
1 2
log

2
2 1

1


k
k  0 2 ( 2 k  1)
= 1 

 (1)
k
k 1
1

2

H Ok

k
k 1 2
(2 k )!!

(2 k  3)!!
J141
( 1) k 1 ( 2 k )!!


k 1 ( 2 k  1)!! k



=


 
=
dx
 ( x  1)
1

 
=
dx
 ( x  1)
0
 /2

 
=

0
.24656042443351259912... 
 
k (2k )

   2 sin
 
k
32 
2  k 1 8

k 1
.2466029308397481445...


dx

cos x  sin x


1 x
 

.2465775113946629846...

1  x2
0
 (2 k )
2
k 1

1
3
3
log 2 
2
2


k 1


(2 k  1)
4k  1
8k 2

2
2
k 1 (8k  1)


Hk
 3 ( k  1)
k 1
k
5
13 (3)
1  ( 2)  2  ( 2)  1  
       


.24664774656563557283... 
144
1728 
648 3
 3
 6 

1
1 .2468083128715153704...  e  e log(e  1)   k

k  0 e ( k  1)
1 .2468502198629158993...  arcsec  
3

.2469158097729950883...

e
k 1
3 .24696970113341457455... 
1
4
30
k2


dx
e
1
1 1 1
x2
log xyz
dx dy dz 
0 0 0 1  xyz
  

log 2 x
1 x3  x 3 dx
.2470062502950185373...
3 .2472180759044068717...

log 3



2
6 3
( 6 )

1


2
k  1 9 k  3k

4

 (k )


3
k
k 2


x
1
3
dx

 x2  x

6
1

4096 30 3870720 k 1 a(k )6
 where a( k ) is the nearest integer to 3 k .
AMM 101, 6, p. 579

3( 4) 



1
.24734587314487935615... 
.24739755275181306469... 

 
=
.2474039592545229296...

.2474724535468611637...

.24777401584773147...

1 .24789678253165704516... 
2 .24789678253165704516... 
1
(  log 4 )   log  ( x ) sin 4x dx 
GR 6.443.1
4
0
1
1  i 3  2  (2)1 / 3  1  i 3  2  (1) 2 / 3 21 / 3  2 2  21 / 3 
2/3
62

1


3
k 2 k  2


1
(1) k
(1) k

sin  Im (1)1 / 12  

2 k 1
k
4
k  0 ( 2k  1)!4
k  0 2 ( 4k  2)!! 4

 7 4
        3 log 2 
 4 3
2
 (k )

2

k
k 2 2  1

( 1) k (12 k )
3
3G 
 
2
2
2
k 1 ( 4 k  1)

1
(1) k k 3

3G   
2
3
2
k 1 ( 4k  1 / 4)



 



.24792943928640702505... 
  (2k ) ( k )  1 
k 2

.24803846578347406657... 

 (k )  1
k 2
3
.24805021344239856140... 
.24813666619074161415... 



125



k 1
2
1
1
 2  4k
k 2
1   1

  2 k   log 1  

j
2k  

coth


1

2

 6k
1

1
2 6
6
k 1
i
i 
1 
1  i 

 i 
 H    H     
.24832930767207351026...     1     1    
2  2
2  2
 2 
 2 

1

 
= 
3
k 1 4k  k

   ( k  1) 
k  1  ( 2 k  1)
 
=  ( 1)


Re

k 
4k
k 1
 k 1 (2i ) 
3
(1) k 1
.24843082992504495613...  8 
 
3
4
k 1 ( k  1 / 2)

2



.24873117470336123904... 
.248754477033784262547... 
.2488053033594882285...

1
  log 2    (22)
12
2

(2)  3( 4)  3(3)  (5)  
.24913159214626539868... 

GR 4.521.3
log A1 , the log of Glaisher’s constant
=
2 .248887103715128928098... 
1 .249367050523975265...
 2 2  1 x 
    2  arcsin x dx 
log
2
 1  2 2  0  x  1

sinh

k3

5
k 1 ( k  1)
2  1  csc h  


sinh 2  csc  

7 4  2


360
6

k 4  2k 2  1
 

4
2
k 1 k  2k  1

2  1  


x4
0 cosh4 x dx 
3
log x
1 e x  1 dx 

.24944135064668743718... 
.2494607332457750217...
  1 1 / 3 
1 
1


1  i 3  1       1  i 3   2  (2) 2 / 3   2 1  2 1 / 3
2/3 


12  2 
2

  2 



 
=
 4k


 (1)

k 1

.2499045736175649326...

 e
3
3
 ( x ) 1

 1 dx



 (3k )
22 k
1
k 1

1
1
( 1) k 1
.2498263975004615315...  sin sinh  

2
2 k 1 ( 4 k  2 )! 2 2 k 1
k 1

1

GR 1.413.1



 .250000000000000000000 =
=
=
=
=
=
=
=
=


1
1
1
 2
 2
4
k 1 3k  6 k
k 1 4 k  4 k

1
J268, K153

k  2 ( k  1)k ( k  1)


1
((2 k )  1)



3
k 2 k  k
k 1

k

4
k 1 4 k  1


2k  1

(1) k 1 (k  1)( (k )  1) 


2
2
k
(
k

1
)
k 2
k 3
k




( 1)
( 1)k
( 1)k 1
( 1)k 1

 2
 2

2
k 2 k  1
k  0 ( k  1)( k  3)
k 1 k  2 k
k 3 k  2 k


 ( 4 k  1)
k

( 1) k
  4

k
4
k 1
k 1 4 k  1



( 1) k 1
( k )
 ( k )2 k





k
k
3k
k 1
k 1 4  1
k 1 4  1




dx
x 3 dx
log x dx
log 2 x dx



1 x5 0 ( x  1)5 1 x 3
1 x 3
1
=
 x arcsin x arccos x dx
0
1
=
  x log x dx
0
 /2
=

 log(sin x ) sin x cos x dx
 /2



0

=
(sin x  x cos x) 2
dx
x5

1  dx
 log1  x  x
3

1

=

0


1 .250000000000000000000 =

2 .250000000000000000000 =

68 .25000000000000000000
=
2
sin x cos x dx
0
0
=
 log sin x 
x 3 dx
ex
4


x 7 dx
ex
4
5
 H2( 2 )
4


FF
9
k
  k 1   k kk 1
4
3
k 1 3
k 1

273
k5
1

  ,  5, 0    k
4
3

k 1 3
Prud. 2.5.29.24
.250204424109600389291... 
  (1) 

.25088497033571728323... 

( k )  1
k 2
1 .25164759779046301759... 

 

1 .25173804073865146774... 
2k  1
lim lg ( n ) pn , p1  2 ; pn  smallest prime  2 p n 1 
n 
Bertrand’s number b such that all floor(2^2^…^b) are prime.
1
 I ( 2)  J 0 ( 2)  
2 0

1
 (2 k )!(2 k )! 
k 0
.2517516771329061417...

1
3 3
1

HypPFQ 1111
, , , ,{ , ,2,2},  

4
2 2
16 
2 .2517525890667211077...

 (10) 
.2518661728632815153...

(( k  1)!) 3

2 
k  1 (( 2 k )!)



4 5
coth
 5
2

1

10

 4k
k 1
1

5
2

 (k )  1 
3
3
1



.251997590741547646964... 
1  
k  k
 (2)  (3)  (4)
k 1
6 .25218484381226192605... 
7 (3)  4 log 2 
 
=
.252235065786835203...

.2523132522020160048...

4 .2523508795026238253...

1 .252504033125214762308... 
.25257519204461276085... 
.2526802514207865349...

1 .2527629684953679957...

.25311355311355311355
=
.25314917581260433564... 
2


2(2 k  1)
 k (2 k  1)
k 1
3
k 2 ( k  2 )

2k
k 1
1
1
dx
 1
1   log 2  log 1     x x

e
e  0 e (e  1)
1

5

2k

I0 (2 2 )  0 F1 (;1; 2)  
2
k  0 ( k !)

1 i  1 i 
 sech  
 

2
 2   2 
 2    (k )
 (3) 
 2 
6
k 1 k
arccsc 4 
 5
Li1  
 7
691
 B6
2730
 2  8 ( 3)


2

3


2 .25317537822610367757... 
.2531758671875729746...
=
.253207692986502289614... 
1 .25322219586794307564... 
.2532310965879049271...

.25326501580752209885... 
k !!

2
)!!
k 1
erf (1) 
1

2
 (k

7
( 1) k 1


k 1 k !( 2 k  1)


1

2  (3)

f (k )

k3

k 1
9 log 3  9   3  2


8
12

3
48

1 .25331413731550025121... 
2


x

8
.25358069953485240581... 
 3k
k 1
3
1
 2k 2
2
cos 2 x dx 
0
  i i log i
 log12
2 3


 /2

1
.25347801172011391904... 
 I (2)  J1 (2) 
4 1

1
1 .25349875569995347164...  
k  2 k ! 1
2 .25356105078342759598... 
Titchmarsh 1.2.15
k
 (2 k )!(2 k )!
k 0

log( x 2  3)
0 x 2  3

1
 (k  3)! k ! 
k 0
.25363533035541760758... 
k 1

1  2
1
2
 5 1   (1) k

 
erfi

1
F
2
,
,




k
 3  1 1 2 2  
1
  e
2
k  0 ( k  2 )!2

1
2 .25374537232763811712... 
24  8e 
e
x1/ 4
dx
0
.25375156554939945296... 
.2537816629587442171...

2 .2539064884185793236...

.25391006493009715683... 
.2539169614342191961...



dx
dx
 4
3

  3x
   4
1
 e 2 x
5
 10 
1 x  x
0 e

5 2  4
1

6

 3(3)  (5)   5
2
6
45
k 1 k ( k  1)

2 2k


2k
k 1 2

1


k2
k 1 4


k
1
1 1 1
log
   Lik     

k
k 1
 2 2
k 2 2
k 1 k 
  

.25392774317526456667... 
1
 (2 k )
k 1


.254023674895814278282... 
2k

1
 k (2k  1) log k
k 1
.2541161907463435341...

1

 1
 ,4,0  Li4   
4

 4

4
k 1
1

k4
k

sin x
dx
ex
1
2
.2543330950302498178... 

4 2  6
2

log 2 log 3
dx
.25435288196373948719... 


  3
3
6
x 1
6 3
1
1  9 11 
5
.25455829718791131122... 
21  erfi (1)  22e  1F1  , ,1 
9 2 2 
32
.254162992999762569536... 



 
  csch 2
 2 coth

16 
2
2


k2
1
= 



2
2
1 2
k 1 ( 2k  1)
k 1 ( 2k  k )
.25457360529167341532... 

 
.2545892393290283910...
1 .2545892393290283910...



CFG D1

1
 k!(2k  7) 
k 0
 
1

e
cosh 1
2
2

1
cosh

2
2
 /2
e
4
 / 2


 (1)
k 1

 4k
k 0
(4k )!16 k


1
1 

(2k  1) 2
k 1 


 

k 1
k (2k )

2k 1


 4k
 (4k )!16
k 1
k

 /2

 
=
 sin x sinh x dx 
0
3 3

4

26 
4

7 .2551974569368714024... 
  log1  3  dx
x  1
3

0
23 .2547075102248651316...

   3k

4 1 

 
3  k  0  2k 

 
k 
e 1
1
2 .25525193041276157045... 
 2 cosh 
2
e
1
5
1
.25541281188299534160... 
log  arctanh 
4
4
2
11 .2551974569368714024...


4
k 0
2 k 1
1

(2k  1)
AS 4.5.64, J941
7
42
 2
( 2 ) 

23
1
 .25555555555555555555 =

90 k 0 (2 k  1)(2 k  7)
1
1
.25567284722879676889... 
(15  8 5arcsinh ) 
25
2

1
1 .25589486222017979491...  

k  0 ( k  2 )! k !
5
.25611766843180047273...   ( 4)  1    
6
4 .25548971297818640064... 


.2561953953354862697...

4 log 2  log 2   
2
2
12


1  2
csch


2
4
2

( 1)k 1
 
= 
2
k 1 2 k  1

7e1/ 3
k2
.2564289918675422335... 

3 
k
3
k 1 ( k  1)! 3

 .25651051462943384518... 
.2562113149146646868...
K133
( 1)k 1 k !(k  1)!
(2 k  1)!
k 1


1
 6    arcsin x log 2 xdx 
0

2
16
3 log 2 2

4

log(1  x 2 )
0 x(1  x) dx
1

60 .25661076956300322279... 

3k k
k 0 k !
3e3  
2

5

H ( 3) k
.2566552446666378985...  
k
k 1 5
1 .2566370614359172954...

2 .2567583341910251478...

1 .25676579620140483035... 
.25696181539935394992... 
4

csc

2
1
 I ( 2)  J 0 ( 2)  
8 0

k2

k  0 ( 2 k )!( 2 k )!

1 .2570142442930860605...

1 1 1

HypPFQ {1},{ , },  

2 2 16 
2 .25702155562579161794...  16 log 2 
53  Hk 3

6 k 1 2 k

1
 2k 
k 0
(2 k )! 
k


.25712309488167048602... 
4
sin 2 x
 3
     
log sin x dx 
2
 4
 5

1
sin
x
0
8  
 4

1 .25727411566918505938... 
.2574778574166709171...


5
 (k  1)

 2 (k  1) 
  2 log 2  log 2 2 
k
k 1
.25766368334716467709... 
2
log 2 2 1  1 

 Li2    
48
8
4  2
( 1)k 1
.25769993632568293344...   3

k 1 k  2
1
log x
 ( x  2)( x  2) dx 
0


1 .25774688694436963001... 
k 2


   ( k ) 

1
k 1
log


k
k 1 k

 

=

log k


2
k 2 k  k

log k

m
m 1 k  2 k

( k )

 ( 1) k

k 1
k 2

H
k
( ( k  1)  1) 
k 1

H ( 3) k


k
k 1 2 ( 2 k  1)
1
I 0 (1)  I1 (1)   1F1  3 , 2,  2  

e
2

.257766634383410892217... 
.2578491922439319876...
1 (2)  2 
   
 5
125
.25791302898862685560... 

.25794569478485536661... 
( 4)  2(3) 
2 .25824371511171325474... 
2 sinh1 

2
(1) k k  2k 
 

k 
k  0 k!2
k 


x 2 dx
0 e2 x  e  3x
22
5 
3
1/( 2 k 1)!

k
k 1
4
1
( k  1)2

k 0
404 .25826782999151312012... 



k

  sinh kkcosh
!
k 0
( k  1)  1

( k )
k 2

.25839365571170619449... 


1 e2
2
e  ee 
4

1
1

Li3     , 3, 0  
 4
4

1
log(1  x / 4) log x
= 
dx 
x
0
 .25846139579657330529... =
 
.2586397057283358176...

2
6
 2 log 2 


k 2

4
k 1
1

k3
k
 ( k )   ( k  1)
2
k 1


k
k 2
2
k 1

(2 k  1)
.258742781782167056...
3 .2587558259772099036...


3 
7
 log 2   
4 
12 
2 
 
=
.25916049726579360505... 

 .259259259259259259259 =
6


0
xe 2 x dx
ex  1

GR 3.42.0

 2 log 2  log 2 2 

0
2 3

2
.25881904510252076235... 

2

log 2
x
dx
2 
ex
1

6 2
CFG C1
2
( 3  1) 
12
4

1
( 1) k
2  2 2arctan


2 k  0 2 k (2 k  3)

7
k2
1

   ,  2, 0    k
27
7

k 1 7
sin

AS 4.3.46, CFG D1





k 


.25938244878246085531...   1 
 2k  
k 1
  
  k 
  


(1) k 
2   1 

3k  2 
k 0 

1 .2599210498948731647...

3
.2599301927099794910...

log8

8
.2602019392137596558...

2


 
=
2
J137
7 (3)

4
2
2

k 1 k  ( k  1)
(1)


2k
k 1


1 .26020571070524171077... 

1 
1 
1 
1  1  

2 
2k  2k 
k 1 2k 
3
 (2 k  1) 
k 1

.26022951451104364006... 
.26039505099275673875... 
.260442806300988445...


 2i
 i e 2i
e  ee
4

1 cos 2
e sin(sin 2) 
2

Bk
log 2 e  1   (1) k

k!2 k k
k 1




sin k cos k

k!
k 1

 3
2  
1
 4

 1  dx
 log
   log log  2

 x x  1
2
 1
0
 
 4
Berndt 5.8.5
GR 4.325.4


 
=
log x dx

x
 e x
e
0

.2605008067101075324...


1
k 

 1  5
k 1

.26051672044433666221... 
 (1) ((k )  1)
k
3

k 2

1 .260591836521356119...


1
1
1

sinh  
2
2 k 1 ( 2 k  1)! 4 k

1
1
 cosh


2 k 0 (2 k )! 2 k
.26054765274687368081... 
.2606009559118338926...

.2606482340867767481...

 
=
.26069875393561620639... 
.2607962396687288864...
.2609764038171073234...


2



 2  3 ( 3)

24
2 arcsin2
1
cot

2k  1

 (2 k )
1
.26116848088744543358...  2 log(cos ) 
2
.26117239648121182407... 


1
1


2
12
3 4 3
3 2 k 1 (3k  1)2

81 log 3
3 2 9 3
1
 ( 4 ) 

 81   5

4
2
2
2
k 1 3k  k

( 1) k 1  ( k  4)


3k
k 1

( k )

k
k 0 5

S 2 ( 2k , k )


k  0 k! S1 ( 2k , k )
csc
k 1

3
 (1)
k 1
J840

k 1
(2 2k  1) B2k

(2 k )! k
AS 4.3.72
( k  1)!( k  1)!
(2 k )!2 k
k 1



2 2
1
1 .26136274645318147699...   3

2
k 1 k  k  k
i 
i 
3 

.26145030431082465654...   ( 3)  3    1 
   1 
 

2 
3
3


 
1 1
 log 1    

p  p 
p prime 


1

= lim   log log n  


p
n  
p prime  n

5

k
742 1/ 3

e 
k
343
k 1 k ! 3
.261497212847642783755... 
4 .2614996683698700833...
 

Mertens constant
.26162407188227391826... 
=
1 .261758473394486609961... 
3 3
1
3
 arctanh  log
4
2
4


 1
1
 1 


Li

Li









k
k
 k 2

 2 
 
k 2 
k 1 4 k ( 2k  1)
log

eG


1 .2617986109848068386...


 1  2
k 1
k
1


(2 k  1) 


 
=
.26179938779914943654... 
arccosh x
1 x 4 dx 

12


 
=
e
3x
0
( 1)


k  0 6k  3



dx
 e 3x

k
4
k 0
2 k 1
 2k 
1
  
(2 k  1)  k 
5
x dx
 1 x
12

0


=
sin x 3  x 3 cos x 3
dx
0
x7
log 2 2 log 3 2 7 ( 3)



12
2
6
8

1
 1
 1
 
= 1  log 2  Li2    Li3     k 3

 2
 2
k  1 2 k ( k  1)
1 .26185950714291487420...  log3 4 
.26182548658308238562...  1 



.26199914181620333205... 

k 1
.2623178579609159738...

.262326598980425730577... 
1 .2624343094110320122...

.2624491973164381282...

.2625896447527351375...

.262600198699885171294... 

1 .2626272556789116834...

 2 (1  log 2)
 log 2 
 ( 4 k  3)
4k  1
16 (3) 
256103

13500
2

1
x 2 log 2 x
0 1  x dx 

k2
 (k  1)  1 
 log     k
4
16
k 1 2 ( k  1)

1
e/2


 iik 
e  / 2  1 1  i i k 0

|  ( k )|( 1) k 1


3k
k 1

( k ) 0 ( k )


2k
k 1

H ( 3) k
2
 4  (3) 1



 
12 180
2
2
k 1 k ( k  1)( k  2)
arctan  
k
k 1 
.2627243708971271954...  
k2
k 2

( k )
.262745097806385042621...   2 k

1
k 1 2
log

.26277559179844674016... 

 
.2629441382869882525...
=

.26294994756616124993... 
.2631123379580001512...


 
=

 
=

.263157894736842105
=
.263189450695716229836... 
2

5
4k  1

 log 2 

2
8
18
k  2 2 k ( 2 k  1)

( k )  1
( 1)k k

2k
k 2

1

1  

k!! 
k 2 

1
7 (3)
x log 2 x
t 2 dt


dx   2t
4
 2t
32
0 1 x
0 e e
1 1 1

 Ei  i 2 ,1  Ei  i 2 ,1   1  i 2   1  i 2 
2 2e 4
1
2  i 2 (i 2 ,0,1)  (i 2 ,0,1) 
4

( 1) k 1

2
k 1 k !( k  2 )
 


2 2

75
 (1)

.263300183272010131698... 
 

 
5
19

p prime
k
1  p 2
1  p 
( k )  1
k 2




.2632337919139127016...

k !!
2 2


 e e dx
x2
0

.26360014128128492209... 
3 .26365763667448738854... 
2 2 3
1
5 1


 2 F1  2,2, ,  
3
27
6
2 4

2
6

.26375471929963096576... 

k
k 1
 (2k  1)
k 
  2k 

H H
5
  k k 2
12
k 1 k ( k  3)

1
1
5
 1
  k
Li2   
2 
 5
4 k  0 5 ( k  1)
4
  (3) 
H ( 2) k

k
k 1 5


.26378417660568080153... 

5 .263789013914324596712... 
.263820220500669832561... 

.2639435073548419286...

 
2
16
4
 log 2 

27
18 9
 
 2k
k 1
3
1

 3k 2
8 2
, the volume of the unit sphere in R5 
15

2
(1) k Bk
[Ramanujan] Berndt Ch. 9
Li2 (e) 
 1  i  
6
k 1 ( k  1)!k

( 1)k 1


2
2
k 1 2 k  k

( k  12 )!
=  

1
k 1 ( k  2 )!( 4 k  1)

 log 2  2 


1


=
 log(1  x
2
) dx 
0

1  dx

2  2
x
 log1  x
0
1

 
=
  arcsin x log x dx 
GR 4.591
0
2 .2639435073548419286...

=

1
1
 log 2        
2
 2
 4
1

1
2 


0 log1  x 2  dx  0 log1  x( x  2)  dx

.26415921800062670474... 
 (1)
k 2

1 .2641811503891615965...

k
k
 (k !)
k 1
3
 

 
 
=
 xe

e
1




( 1) k
( 1) k 1 k
( 1) k k 4




k  0 k !( k  2 )
k 1 ( k  1)!
k  0 ( k  1)!


( 1)k
1
= 


k 1 ( 2 k )!( k  1)
k  0 ( k  1)!  k !
2
n


1 
k log k
k k Hk
=  ( 1)

   ( 1)
k!
k!
k 1
n 1 n ! k  2
2
.26424111765711535681...  1  
e

( k ) k

2k
x
log x
dx 
x2
1
dx  
0
.264247851441806613...

.264261279935529928...
( 3  1)

1 .2644997803484442092...

3 1
3 1
log

16
4
2

2
7
x2
 
csc
 5

25
5
( x  1) 2
0



1


k
k 0 2  1


k 1
 ( k )  ( 1) k
2k  1


 (1)k
k 0
(2  3 )2k 1

4k  1

.2645364561314071182...

.26516504294495532165... 
121 
1

5 k 0 k !( k  1)( k  6)

3
( 1) k 1 k 2  2 k 
  k
 
8 2
k  1 4 ( 2 k  1)  k 
9e 

( k )

F
4 .26531129233226693864... 
k 2
k 1
4 sin 1
.26549889859872509761... 

17  8 cos1
1 .26551212348464539649... 
4744 .2655508035287003564...

1 .2656250596046447754...

sin k

k
k 1 4

 1 
log 2   log       
 2

8
2
1
2
k!

1
k 
4  1 k 1 ( 4 )3
k 0


k



3
1

 (e2  1)  4e  4e cosh1  (2  2e2 )( si (1)  ci (1)) 
2e


(1  e2 ) log 2  (e2  1)ci(2)  (e2  1)si(2) 




 ( 3k )
k 1
1 .2656974916168336867...



.2656288146972656805...

1
1
log 2  2 Ei (1)  Ei (2)

 2   sinh 1  eEi(1)   eEi(2) 
 e log 2 
e
2
e


Hk
 
= 

k 1 ( 2 k  1)!
1
2
 2 2 
.26580222883407969212...  sech2 
cosh 2 e  e
 
=
.265827997639478656858...
1  
H 1 i
2 5  




 
e



5  1 
 H 1  i

2 




5  1 
1  5 
1  5  

 H 1 
 H 1 


2 
2 
2 






=
 F  (2k  1)  1 
k 1

k
2G 
 log(2  3 ) 
3 12

1
.2658700952308663684...   k ( k 1) 
k 1 2

1 1
ke

4 .2660590852787117341... 
 ,  e,1   k
e e

k 1 e
1 .2660658777520083356...  I o (1) 
.2658649582793069827...

Berndt 9.18


 
J (1)
1

=  k
e 
k  0 k!
(k  12 )!2k


(k!) 2
k 0

1

 
=
e
cos x
dx
0
.26607684664517036909... 
9 1  (1)  5
 1 
      (1)    
 6
 3 
4 4
( 1) k 

2 
k  1 ( k  2 / 3)

31 6 465 (6)
dx


   (log5 x )
252
4
1 x
0
1
118 .2661309556922124918...



x 5 dx
0 1  e x 
.2662553420414154886...

 cos(  2 ) 
1 .2663099938221493948...




( 1)k k 2

Fk
k 1


5 .26636677634645847824... 

k 2

2 .26653450769984883507... 
2
(k ) 
dx
 ( x) 
1
32 5
log 4 x
dx
  2
x
x


1
243 3
0
1
.26665285034506621240... 
  3 coth  csch 2  i  (3, 2  i )   (3, 2  i )  
2

x 2 sin x
 
=  x x
dx 
0 e e 1

|  ( 2 k )|
.26666285196940009798...  
4k
k 1
1
23 .266574141643676834793... 


.266666666666666666666 =
1 .266741049904...

1 .26677747056299951115... 
4

15


k 1



 e
dx
x1 / 4
0
1

1
 1  k (k  1)(k  2)  



1  log(e  1)
log x
 
dx
2
e 1
x
e
(


1
)
0
1
.26693825063518343798... 
2 .266991747212676201917... 
 2 k  1
1

k 
k 
k  0 (k + 1)8 

8 2 log 2  log(2  2 )  
11 e

8

k3
 k !2
k 1
k

GR 4.264.1
.2673999983697851853...

 

 

 


1
log 2  2 
2



1
= log cos  sin   log

2
8
8

1  k 
=  cos

 4
k 1 k

=




2 2  2 2  

 1 i
    ( 1)1/ 4 2  log(1  ( 1)1/ 4 )  log(1  ( 1) 3/ 4 ) 
 4 4

1
 3
Li2   
 2log 3  2 log 2  log 2  Li2   
6
 4
 4
1

( 1) k 1 Hk
log(1  x / 4)
 
dx 

k
3 k
x
k 1
0
2
.26765263908273260692... 

 
=

5 .267778605597073091897... 
2 log 2  7 (3) 
2

0
( 1) 2

.267793401721690887137... 
k
x 2 sin x dx
1  cos x
k
 ( k  1)! (2 k )
Titchmarsh 14.32.1
k 1
 2k 
1
 
(k  1)  k 
k 1

 2k 
1
  
1 .26794919243112270647...  3  3   k
k  0 6 ( k  1)  k 
.26794919243112270647... 
2  3

6
k

4 .2681148649088081629...


.268253843893107859...

160 8 3 64


log 2 
27
27
27
6

4

2

x
.26903975345638420957... 
4e
Li2 (  x ) 2 dx
0

f3 ( k )

2
k 1 k
 1  ((2)  1)3  
216 12 2

1
( 1) k 1
.268941421369995120749... 
 
e 1
ek
k 1
 erfi 1
 5/ 2
Titchmarsh 1.2.14

  e  x sin x cos xdx
2
0

k k
.269205039384214394948...   ( 1)
Fk
k 1
( 1) k 1 k
 

2
k 1 k  1

.2696105027080089818...


 
=

1
cos x
cos(log x )
0 1  e x dx  0 1  x dx 
i
i
1 
 1 i 

1 i
 1     1              


 2 2
 2 2 
4 
2
2



 
4 .2698671113367835...
.270034797849637221...
=

1 i 
 i
1 i 
 1  i 
        
  
 

4   2
 2
 2 
 2 
2

 
=
GR 3.973.4
cos xdx

2
x

2
0

2
2

dx

x


3 (3)

sin(sin x  x)
cos x
0

2e
.27007167349302089432... 
e

2



e
 2 log   2 log 2  12 log(Glaisher )   
22

3
 k  4  (k  1)  1
k
k 1

16 

105

cos1  ( 1) k
( 1) k k
.27015115293406985870... 



2
k  0 2 ( 2 k )!
k 1 ( 2 k  1)!( 2 k  1)
.27008820585226910892... 
.27020975013529207772... 
.270222741100238847013... 

.27027668478811225897... 
.270310072072109588...


2
1
1
x dx
arctan   4

8
2
4
x 1
1

6 (3)
11
k
 (2k )  1

2
log




2
4

k 1 k  2


2
log 2
log x


dx 
16
2
( x  1) 2 ( x  1)
1
2
3
log 
3
2

 (1)
k 1
k 1
Hk

2k
J 137

.2703628454614781700...

 
1 .2703628454614781700...
( 1)k
(( k )  1) 
k
k 2

1 
1
=    log


11/ k 
k 2 k


log 2    1  
log 2   
2

 

 H ( x) dx 
1


 
=

1
  1  x
0
.2704365764717983238...
 dx
 cos x 
 x


1

4 2
2 k 1 4 k  4 k  1

5
1  2 k  2
(3 5  5)   k 

k 
2
k 0 5 
 1 
4 .27050983124842272307... 

2
tan

2
K 124h
.27051653806731302836... 
.270580808427784548...

.270670566473225383788... 
3
2 3
1 x
log
  log
dx
2
1  x6
2 3
0
1
5    log 2 
4
360

2

e2
 (4)

Hk
 MHS (3,1) 
3
k 1 ( k  1)

4
(1) k 1 2k


k 1 ( k  1)!
(1) k k 2 2k


k!
k 1


(1) k k 3 2k


k!
k 1


3 .27072717208067888495... 
1 .27074704126839914207... 
1 .2709398238358032492...


 .27101495139941834789... 
2 log 2 
 3 log 2

12
3
e

2 e  1)
(1) Bk
2 k k!
k 0
2



 4G 
2
k
2
k 1
3 (3)


2
 /2

0
x4
dx 
sin 4 x
k
 
k


5
4
  k  1, 

1  1 
    i    i
2  2 
cosh 

=
3
 (1)
k 1
22 k  (2k  1)
k 1
.27103467023440152719... 

 .2711198115330838692...


 
=
.27122707202423443856... 
1 .27128710490414662707... 
.271360410318151750467... 
.27154031740762188924... 
.2716916193741975876...

1 .2717234563121371107...


 ( 3  1)
6 2

3 2



 
3 1

0
x4
dx 
x12  1


0
x6
dx 
x12  1
log 2 1   1
i 
i 
i 
i 
1


   
   
   1 
   1 
 
2
8 2
2
2
2  
2
2



( 1) k 1

3
k 1 k  2 k

e
k
1
1
1

 sinh  

k
16 16 e 4
2
k 1 ( 2 k  1)! 4

(1) 3 / 4   5 
1
1

1 5 

 4     ,  1   1 F1  , ,1  

4  4
4

4 4 
k  0 k!( 4k  1)

1
1
k
sinh
 
k
2 2
2
k  1 ( 2 k )! 2

cosh 11
1
 sinh 2 
2
2
2

 1  dx
2  3
x
 sinh x
1
 /2
2
12
 log sin x
 2 log 2  log 2 
2
 1
0 F1  ;2;   2 I1 ( 2 ) 
 2
0

2
k 0
k
1
k !( k  1)!
2
sin x dx 
63  5 3  45 log 3 
1


150
k 1 k (3k  5)
e
.2718281828549045235... 

10
.2717962498229888776...



1 .27201964595140689642... 

.27202905498213316295... 

 
=

 
=


1 .2721655269759086776...

.27219826128795026631... 



1 
2 


1




1  2  


2
sinh 
k  1
k  1
k 1 
k 2 
 3  i   3  i 
 (i )  ( i ) 
  (1  i )  (1  i ) 
10
1
(2  i )(2  i )  10(2  i )(2  i ) 
2
6

9
 log 2
i  1 i 
 1  i 
 G   Li2 
  Li2 
  
8
2  2 
 2 


log(1  x)
dx 
1  x2
0
1

=

GR 4.291.8

= 
log( x  1)
dx 
1  x2
1

1
=
log x dx

1  x4
0
log 2
=

 log(sin x)

0
sin x dx
1  sin 2 x
 1  x  dx


x  1  x2
1
 log
0
 /4
=
GR 4.243
x dx

e  2e  x  2
 /2
=

GR 3.418.3
x
0
=
GR 4.291.11

0

GR 4.386.1
GR 4.297.3
x dx

(cos x  sin x ) cos x
1
=
  x arccos x log x dx 
0
1

 
=
arctan x
dx
x 1
0

 /4
=
 log(1  tan x) dx 
0
GR 4.227.9
 /4
=
 log((cot x)  1) dx 
0

.272204279827698971...

k
k 2

4
GR 4.227.14
1

 12

15 .272386255090690069056... 
5 
1

4 log 2    k
2 k 1 2 ( k  2 )

1
1
6 .2726176910987667721...  8  2 cot
 2 2

1
4
k 1 k   16
1
.27270887912132973503...  3  2 e    log 2  Ei ( ) 
2
3
 .27272727272727272727 =
11

1
(1)
.27292258601186271514...   (1   )  

2
k 1 (   k )
.2725887222397812377...

2k

k 1 Fk Fk

1 .27312531205531787229... 
.273167869100517867...


 5


k
 ( k  1)! 2
k 1
k2


4
2
2
2 5
k  2 k  3k  1

2
1
1

arcsin

7
2 2 k 1 k  2 k 
2  k
k
1
tan

k
k


 F  (2k )  1 
k 1
2k

 1 
 ( 2)
5
 (2 k  3)!!
1 .2732395447351626862... 

 
J274
 
 
2  
 1 / 2

 ( 3 / 2)
4 k  2  (2 k )!! 

 2k 
1
=    k
k  0  k  16 ( k  1)

1

=  k tan k  2
K ex. 105
2
k 0 2

( 2 k  1) 2
= 
k 1 2 k ( 2 k  2 )

1
.2733618190819584304...   k  2

1
k 1 2
26 (3)
4 3
1
 2

   ( 2)  
.27351246536656649216... 
27
27
81 3
 3
2
4

=

x2
log 2 x
1 x3  1 dx  1 e2 x  e x dx 

9 .2736184954957037525...

86
1
3
dx
2
1 .27380620491960053093...   log   log(1  3x )

2 0
1  x2

1
H
.2738423195924228646... 

8 log 2 2  14 log 2  13   k k

2
k  1 2 ( k  3)
1
54  3 3   2  27 log 3  2 (3) 
.27389166654145381...

2


1
( k  3)
 
=  4

( 1)k 1

3
3k
k 1 3k  k
k 1


.2738982192085186132...

GR 4.295.38



 
 
  
1 
    2 arctanh  sin   cos     2 arctanh  cos   sin  

8 
8 
8 
8 
8 




 
=
x
1

.273957549172835462688... 

k 2
4
dx
 x4
 (k )  1
2k  1


1
1
 1
arctanh
 2 
k
k k 
k 2
 


 2 sin  3
3 

  1 

( k  2) 2 
 3
k 1 
k

 cos
1
2
3 
.2741556778080377394... 
 
 
2
2
k
36
k

k 1 
k 1
2   k 2
k 
.27413352840916617145... 



 
2 .2741699796952078083...
1 .2743205342359344929...
28 .2743338230813914616...
1  dx
log(1  x 6 )

=  log1  3 
dx
 
x x
x

1
0

1  2 k  3
 32(5 2  7)   k 

k 
k 0 8 

1



coth
1   2

2
2
k 1 ( k  1 / 2)

k 1  ( 2k )
=  ( 1)
2k 1
k 1

.27443271527712032311... 
1
9
3 1
 2 4 5 log 
6
1



 2 F1  2,2, ,  
2 4
25
6
125
2

J274


 
=
 (1)
k 1
k 1
k

 2k 
 
k 

83711
1

304920 k 1 k ( k  11)

1 1
k
 1
.27454031009933117475...   k  Li1/ 2    ( , ,0) 
 5
5 2
k 1 5
.2745343040797586252...


.2746530721670274228...

 
log 3
1
1

 arctanh

4
2
2

dx
=  2 x

e 2
0

.2747164641260063488...

 ( 1)
k 1
9
7452 .274833361552916627...



k 1

4
k 1
k
1

( 2 k  1)
H2 k 1 
5
k     2 arctan 2  2 log  
4
5
4
4

4 
1
.27489603948279800810... 

  log 2  
9 6
k 1 k ( 4 k  3)

1
1 .27498151558114209935...   k

k
k 1 3  2

 .275000000000000000000 =
11
40

.27509195393450010741... 
sin k
 ( k  1)
k 1
3 .27523219111171830436... 
.27538770702495781532... 
2

i i
e  Li 2 (ei )  e 2i Li 2 (e  i )  
2
3

G
2
12
 log 2 
log 2 2
 1
 1  log 2  Li2    1 
 2
2

1
k ( k  1)
k 1

( 1) k 1

.27539847458111520468...  
k  1 k ! ( 2 k )

( 1)k
.27543695159824347151...   3

3
k 0 k  k  k  1

 



 

=
2
k
2
log 2 1 
i
i
1
1 i
1 i 
 2 csch 
  (1  )   (1  )   (  )   (  ) 
4
2
8
2
2
2 2
2 2 
i

.27557534443399966272...   
 log (1  i )  log (1  i )  
2 2


1
( 1) k  1 (2 k  1)
1
 
=    arctan   

(2 k  1)
k  k 1
k 1 k
=
1
.27568727380043716389... 
  log x tan x dx 
0
1
1  2 2k (2 2k 1  1)B2k

J132, AS 4.5.65


e 2  e 2 2 k 1
(2 k )!
log(cos 1) log(  cos 2 ) 
cos 4 k
.27574430680044962269...  log 2 

  ( 1) k 1
2
8
k
k 1
2
1
G  (3) 


log(1  i )  Li3 (i ) 
.275806840982172055143... 
4
2
16
2
 /4
1
2
 
=
16 G  2 log 2  35 (3)   x 2 cot x dx
64
0
.27572056477178320776... 

csch 2 


3 .275822918721811...

Levy constant
4 .2758373284623804537...

8
5
2
4
4


csc
5
5
5 5

.27591672059822730077... 

1 




0
 (k )
 (1) 1   (k )    k (k  1)
k
k 2
k 2
x dx
 1 x
5/ 2
4
.27613216654423932362... 

.27614994701951815422... 
.27625744117189995523... 
1


csch  2 
4 2 2
(1) k 1

2
k 1 k  2 k  3



1
1

  2

8 12 3
k 1 9k  4

5 3
log 2 x
dx 
  3
x  x3
324 3
0

13
1 .27627201552085350313... 

32
x 3  sin 3 x
0 x5 dx 
GR 3.787.3
2 
(1) k
1
Erf   


 k  0 k!(42 k 1 )(2k  1)
 4

 ( 3)

1  (1   )
1
.2763586311608143417... 
 3  
 3
3



6
k 1 k ( k   )

 2k 
1
5
.27648051389327864275...  log 2 
   k

12 k 1  k  4 k ( k  2)
.276326390168236933...


1 .27650248066272008091... 
.27657738541362436498... 
2 .27660348762875491939... 
1 .276631920058325092360... 
2
( 1) k 1
4 
8   2 2

3
k 1 k ( k  1 / 4)

( 1)k 1


k 1 k !  1

ek
2
I2 (2 e )  0 F1 (; 3; e)  2 
e
k  0 k !( k  2 )!

  1  (1) k ( k 1) / 2 
1 3 1
1
 


 ,   
8  2 4  k  0 (4k  1)3 / 2
2  2 0 k  k  2 

[Ramanujan] Berndt IV p. 405
k 1
(1)

k  1  3k 
 
k

.276837783997013443598... 

.276877988824579262196... 
4 log 2 2 
.27700075399818548102...  

2
6


Hk
 k (k  1)(2k  1) 
k 1
43
9  13

csc  13 
234
234

( 1)k

2
k  4 k  13
2  


2
=
.2770211831173581170...
( 1) k 1 ( k  12 )! (   1) k

( k  1)!  k
k 1

.27704798770564582706... 
14
1
8 arctan 2  3 log 5  56  

25 100

.27708998545725868233... 

 xe
x
log x sin 2 x dx
0
1
 4 ((k )  1) 
k
k 2

.2771173658778438152...

 ((k )  1) 
3
k 2
4 .27714550058094978113... 
.27723885095508739363... 
1 1
  , 2, 
 2 2
4
3
 3
1
  Li3   
 4
 4
k 1


( 1) H (2) k
H
=  k k 2  

3k k
k 1 4 k
k 1

 


 



 .27732500588274005705... 
  Li3 
 log 2 2 
 1
  
 3

x log(1  x )
1 .27739019782838851219...  
dx 
ex  1
0
.27734304784012952697... 
1 .277409057559636731195... 
 3
12

3log 3

4
 
=

 3k
k 1
2
1

 2k
log(1  x )
dx 
3
x
0
1

3

(1) k Bk
 Li2 (1  e)  1  

k 1 ( k  1)!

.27750463411224827642... 

(1) k Bk

 Li2 (1  e)  
k  0 ( k  1)!

1 .27750463411224827642... 

 .2776801836348978904...

192 .27783871506088740604... 


dx
  2

( x  2) 2
8 2
0
6


x 2 dx
0 ( x 4  1) 2 
5
i
i
1 .27792255262726960230...  2 sin log 2   i ( 2  2 ) 

.2779871641507590436...


 3
 4
 (3)  log 2 log 2 3  4 log 3 log 2 2  4 log 2  log Li2   
log 7

7

1 .2779927307151103419...

 (k
k 2
2
1k
 k ) log 2 k
5 (3)
2
 2 log 2  log 2 2 
 4G 
8
6

( 1) k 1 Hk
 
=  2

k  1 k ( 2 k  1)
arcsinh2  ( 1) k 1  2 k 
.27818226241059482875...  1 

 
2
k 1 2 k  1  k 


 (2 k )
1  2
  I1  
2 .27820645668385604647...  
k
k  1 k !( k  1)!
k 1 k
.278114315443049617352... 

1
log 2 x

    4, 3,    2

2  0 x  1/ 4

1
7 .278557014709636999...

 
=
.27865247955551829632... 
3
 i
 i 
  log2 2  2i  Li3    Li3     
 2
  2
4
1
1

2 log 2
11
17 .27875959474386281154... 

2
.2788055852806619765...

.27883951715812842167... 
.27892943914276219471... 

1


k
k 1 2

dx
2
x

1

1 
 (1  erf 1)   , 1 
2 
e
5

2 
2 2e

e
x
1
 1  dx
 sinh x  x
dx
x

4
1
5
5  H
1  1
log   kk   k 1
4
4 k 1 5
4 k 1 5 k

2 .27899152293407918396... 
 k log (k ) 
k 2
.27918783603518461481... 


 .27930095364867322411... 
 ( k )   ( k  1)
k 2
k

3
 3


 6k
2
k 0

S2 (2 k , k )
.27937855536096096724...  

2k
k 1 ( 2 k )
.27937884849256930308... 


 
=

k  1
1

 
2  1  k log 1 


k 
k 1 k


1

2
1/ 6
1 1 
 1 3 
  ,1    ,   
 2 2
2 2 
1
 1 
 2  ( 2  2 )    
 2 
2
( 1) k

 3 (2 k  3) 
k 0


k 1
k
( 1) k 1

2k  2

1
( 1)k 1




k
k
k 1 4  1
k 1 4  1

.2794002624059601442...


 .27956707220948995874... 

1
( 1) k 1 k 2  2 k 
J0 ( 2 )  

k 
2
k
k  0 ( 2 k )! 2

2 .2795853023360672674...
1

2
k  0 ( k!)

I 0 (2)  
=
0 F1 (;1;1) 
1
2
2
e

k2


2
k  0 ( k!)
2 cos 
LY 6.112
d
Marsden p. 203
0
1  2k 
  

k  0 ( 2 k )!  k 

=


k 2  2k 
( 1) k
2
  e 

k!
k  0 ( 2 k )!  k 
k 0

 2k 
 
k

1
1
 .2797954925972408684... 

2   (1  i 5 )   (1  i 5 )   3
10
k  1 k  5k

i i
sin k
i
cos 1
1 .2798830013730224939...  e

sin(sin 1)   ee  ee  
2
k 1 k !



1 .27997614913081785...
.2800000000000000000
2 erfi( 2 )

e2

=
.28010112968305596106... 
7

25

4e

p prime
( 1) k 18k

 2k 
k 1
k ! 
k
1  p 
( e  1) 
2 3

0
GR 1.449.1

1  p 6


sin 2 x
ex
2

dx

7 .2801098892805182711...
53 


.28015988490015307012... 
.2802481474936587141...


(k )
k 1
 
k
  2k  

1 
3
3 
   1  i     1  i   
3
6 
2
2 



 2k
k 1
3
1

 3k
1
2 
1
  3  9 log 3  2 (1) ( )  
2 
6
3 
k  1 k ( 3k  1)

5
1
 3
.28037230554677604783... 

 2 log 2  hg     2
 2
3
k 2 2k  k

( 1) k ( ( k )  1)
 
= 

2 k 1
k 2
.2802730522345168582...



8
3 
1
 2 log 2 


3
2 k  2 k ( k  32 )
1 (1)  1 
1
   

.28043325348408594672... 
 3 k  1 ( 6k  4) 2
36

1
1 1 1

1 .28049886910873569104...  
, , },{ , }, 
2  HypPFQ {111

2 2 16 
k  0  2k 
 
k
i
.280536386394339161571...  (1   ) sin 1 
log  2  e  i  log  2  ei 
2

sin k
 (k )  1
 
= 
k 2 k
1 .28037230554677604783... 



6 .28063130354983230561... 
 k (
2




( k )  1) 
k 2
 1
2  log 2  log 2 2  l    
 2

352 2 log 2
1
.28086951303729453745... 

 
735
7
k 1 k ( 2k  7 )
.2806438353212647928...

Berndt 8.17.8
e  1  (1  2k ) (1  k )

.28092980362016137146...  log

2
k!2 k
k 1
(1) k (2k  1) Bk
 
= 
[Ramanujan] Berndt Ch. 5
k !2 k k
k 1

64
1 ( 2)  3
64
1
3
.2809462377984494453...  
     28 (3)   
 

3 3
 4
27 2
27
k 1 ( k  4 )

1
 2 25
.2810251833787232758... 

 3
24 192 k 1 k  4 k 2

 3 1 
( 1) k 1 2 k
 2  

.28103798890283904259...  2 3 log
2
 2
k 1  2 k 
  (2 k  1) k
k


2 .281037988902839042593... 


3 log
31

31

( 1) k 1 cos k
1
1

.28128147005811129632... 
log ( 2 cos )  
2
2
k
k 1


k
1
.28171817154095476464...  3  e  

 
k  0 ( k  2)!
k  2 k!k ( k  1)


k
1
 
= 


k  0 k !( 2 k  6 )
k  0 k !( k  2 )( k  3)
GR 1.441.4

k2
1 .28171817154095476464...  4  e  

k  0 k!( k  2)
.28173321065145428066... 
.28184380823877678730... 

 
=
.281864748315611781912... 

e  1 cos(sin 2)  ecos 2
(1) k 1 sin 2 k

 

2e
2ecos 2
k!
k 1
7 
3 
  3


  cot
 cot
  log 2  2  log sin  log sin  
4
8
8 
8
8 

k 1

( 1)

2
k 1 4 k  k
1
2 log 2  2     1 , 1     1 , 3   
8
 2 4 
  2 4
( 1) k 1 log(2 k  1)


2k  1
k 1

3k 1
2
 log 2 
  2

2
24
k 1 2 k ( 2 k 1)

( 1)k 1 k( k )
= 

2k
k 3


 
.2819136638478887003...

 

 
=
1
=
 arctanh x log x dx
0
1
4
.28209479177387814347... 
.28230880383984003308... 
( 1) k 1 H (3) k


3k k
k 1
.282352941176470588
24
sin 4 x
 x 
85 0 e


=

1
exp    (1)  , Glaisher’s constant

 12

( 1) k
1 .28251500934298169528...   3  6 log 2  
2
k  0 ( k  1)( k  3 )
 2 log 2 8 log 3 2
 1
.28253095179252723739... 

 2 log 2 2 log 3  4 log 2 Li2    3 (3) 
 4
3
3
 1
 
  4 Li3   
 4
1 .282427129100622636875... 

log 2 (1  x )
= 
dx 
x ( x  12 )
0
1

 
1 .2825498301618640955...


6

( 2) 
.2825795519962425136...

.28267861238222538196... 
.282698133451692176803... 

 
.2827674723312838187...
=

.282823346997933107054... 

k
1

I1 (1)  
2 k
2
k  0 ( k !) 4

1


3
k 1 k  6

1  2
 2 
   2 H   , 2    3 log 3  1  9 log 3 
12 
 3 


Hk


k 1 (3k  1)(3k  1)

1


3
k 2 k  3

1

k 1  2k 
  k (k  1)
k 
.2829799868805425028...

0 F1 ( ; 2;  2) 
.2831421008321609417...

( k )  1

k  2 k ( k )
.28318530717958648...


J1 ( 2 2 )
(1) k 2k


2
k  0 k!( k  1)!


1
2  6   arcsin 2 x arccos x dx 
0
6 .28318530717958648...

 
 1  5
2        
 6  6

1
= 

1
3
k  0 ( k  4 )( k  4 )


=
 log1  x  dx 
6
0
2

 
=

0

6 .2832291305689209135...

2 .28333333333333333333
.2833796840775488247...
.2834686894262107496...
d

2  cos

e x / 6dx
 ex  1 

2  coth 2  
137
 H5 
=
60


 (2 k )
1


 
   k sinh  1 


k
k  1 ( 2 k  1)!
k 1
k 1

( 1)
1/ 3
 1  e


k
k 1 k ! 3


5 .28350800118212351862... 
2
3/ 4

 log1  2 x  dx 
4
0
3

.28360467567550685407...  3 log 2  3 log 3 
2


k


k
k 1 3 ( k  1)
(1) k


k
k  0 2 ( k  1)( k  3)

 (k )

1 1
 


 log 1     

k
 2k  2k 
2
k
k 2
k 1 

6
6
k Hk
.28375717363054911029... 
 1  log    k 
25 
5  k 1 6
2

49
1

.28382295573711532536... 

  2   (1) (4)   (2,4)   1, 2, 4  
6
36
k 4 k
.2837571104739336568...

log  
.2838338208091531730...

1
1
1 2  
4 e 
1 .28402541668774148073... 


k 1


 
=
.28422698551241120133... 
( 1) k 2 k


k  0 ( k  2 )!

(2 k )  (2 k  2)
( 2 k )!
 1
  k
k 2

2
1
sinh x
0 e x dx 
1
1
 1  coth x dx 
0
e
4

.28403142497970661726... 


2

1
1

 1   1  2  cosh  

k 
k
ci(1)sin 1  ci( 2 )sin 1  si(1) cos 1  si( 2 ) cos(1) 
1

 
6 .2842701641260997176...
sin x dx

1 x
0

k Hk 2
 
k
k 1 2
=

2

log 2 3 log2 2 log 3
 1
 log 2 

 Li2    log 2 
.28427535596883239397... 
 2
12
6
4
1  1  21
 (3) 
 

  Li3    
2  2  48
2
log 2 (1  x )
0 x 2 ( x  2) dx 
2 .28438013687073247692...  (3)  ( 4)
1

 
=

.28439044848526304562... 
 (k )
 (2k  1)!
k 2
8

11

Fk
3 .28492699492757736265...  
k  1 k!!
 2 log 2
.28504535266071318937... 
24
2 .28479465715621326434... 

 1
1 1
  
sinh
k
k k
k 1

 

1 .28515906306284057434... 
1
 k ! (2k )
k 1
.28539816339744830962... 
=

 2 2
.285185277995789630197... 
2


9/4
e
 x2
sin x 2 dx 
0
( 1) k 1
1
  2 
4 2
k 1 4 k  1


sin k cos k
sin k cos2 k



k
k
k 1
k 1


J366, J606
1
=
 x arctan x dx 
0
.2855993321445266580...



11
15e
1
1 .2856908396267850266... 


32
32e
2
.285714285714285714
=
7
1 . 2857644965889154964... 
.285816417082407503426... 

k4


k  0 ( 2k )!


5 7
2   2  
8 8
 (3)
3

2


4
k 0
11

162
(4k )!

( k! ) 4
4k

k
54
k 1
2
k 1 k
1 .2858945918269595525...   ( 1)

k !!
k 1
 2  2  3 log 2 G
.28602878170071023341... 



16
4
2

k ( k )
 
= 

4k
k 2
3
1

(k  3)



.2860913089823752704...

.28623351671205660912... 
.2866116523516815594...
.2868382595494097246...
(3)  G 
2 1



8k  1
 4 k (4 k  1)
k 1
2

(1) k 1


3
2
k 1 k  2k

24 8
6 2


16
1
1
1 
( k !) 2

HypPFQ {1111
, , , },{ ,2,2},   


2
4  k  0 (2 k )!(2 k  2) 2
4
.286924988361124608425... 

4 2
coth

2

2
8
csch
2

2
 
2 9

2k 2


2
k  2 ( 2 k  1)
CFG D1


 
=
 ( 1)
k 1
k
 (2 k )  1
2k
k 1
1 .28701743034606835519... 


1   1
 

e1/ 32  1 
erf
2
4 2  k 1 k !! 4 k


1 .287044548647559922...


1 
2

 1  ( k !)
k 2

1 .287159051356654205862... 
  e
( k ) 1
k 2
3
1 .2872818803541853045...
.28735299049400502...

 1 
2e  17  3k k 2


18
k 1 ( k  3)!
 c4 

3 .2876820724517809274...
 
i
dx
=  2

1 x  x

k 1

dx
1 3x 2  x 
log 2

0

1

k
k
4
k 1

dx
dx

 x
x
e  1 0 3e  1

(1)
1
1
1
Li
2


 2arctanh 




1
k
2 k 1
(2k  1)
7
 4
k 1 3 k
k 0 7
e

k
10 .287788753390262846...
 

k 0 k !
 1
1 .28802252469807745737...  log   
 4
2 .288037795340032418...
  (  ) 
6
1
.2881757683093445651...        hg   
5
5
2
5 log 5
5

2
1


 
= 5 

log
2
4
4
5
3 5


 (k  1)
1
 
=  2

  (1) k 1
5k
k 1
k 1 5k  k

t (k )
3 .28836238184562492552...   3
k 1 k!


i
2 ecos 1 cos(sin 1)  ee  ee
1

 2 log 2  log 3   ,1,1 
4

2

Patterson Ex. 4.4.2
1
(  5  10  3(2)  20  2(3)  15  2 (2)  30  ( 4)  24(5)) 
  
120
k
k /2
sin
 2
4
2 .2873552871788423912...  e sin 1  
k
!
k 1
 1 i 
 log 2
1  i 
 i  Li2 
1 .287534665778537497484...  2G 
  Li2 
 
4
 2 
  2 
2 .28767128758328065181... 


 
=
J117
 (k )
  (k )


 1 
(k  1)
k 2
k 2
  (k  1)


1
i  2 i
 2 i 

           
 3 3  k  0 (3k  2) 2  1
6   3 3

.288385435416730381149... 
.28852888971289910742... 
.2886078324507664303...

k

 
=
2
 (e
x2
 e x )
0

  (e  x 
2
=
0

=
.2887880950866024213...
 (1)
k
dx

x
GR 3.463
dx
1
) 
x 1 x
GR 3.467
2
e  x  cos x
dx 
x
2

0
.2886294361119890619...


2



log 2
3



5
20

1

   1  k  
2 
k 1 

1
 (2k  1)(2k  6) 
k 0

1
1


1   (1) k  ( 3k 2  k ) / 2  ( 3k 2  k ) / 2 
2
2

k 1

log  (k )
.28881854182630927207...  
k  2 k ( k  1)
=
1 .288863135...

.28893183744773042948... 


1
1  2

2 
p ( p  1) 
p prime 

 /2

4e
  sin(tan x ) sin 2 x tan x dx 

1
 4 (2k  1) 
k
k 1
.2890254822222362424...
.289025491920818...



sin  x
dx 
  1 0 ex
1


one  ninth constant

GR 3.716.8
0

.2889466641286552456...
Hall Thm. 4.1.3
2


4 
( 1) k
( k !) 2 3k




2
3 k  0 3k ( 2 k  1) 2
k  0 ( 2 k )!( 2 k  1)


10  2
1
= 
log 3 
 5

2
3 3
27
k  0 (3k  1)
10 2 5 (1)  1 

= 
log 3 
   
 3
27
9
3 3
GR 3.463
1 .28903527533251028409... 

 
Berndt 32.7


1  1 1
1
 3
  , 2,  = HypPFQ  1,1,1,1,  ,2,2,  
3  3 2
2
 4

 1
5 .28903989659218829555...     
 5
 
=
( 1) k

2
k  2 k log k

( 1)k 1
.289144648570671583112...  

Fk
k 1
2 3  3

.28922492052927723132... 
48

.2891098726196906...


2 log 
1001
.289725036179450072359... 

3455
2 .2894597716988003483...

2 .2898200986307827102...

.2898681336964528729...

 
=
2
3
3 


1  p 
2 2
p prime

1


2
2
k 1 k ( k  1)


k
 (k  1)(k  2)
k 1
2

K Ex. 111
 k ((k  3)  1)
k 1
2
1 .2898681336964528729...
1  p  2  p 4
li(  )


( 1)k


k  0 ( k  3)(3k  1)


3
 2  2 (2)  2 

 (k (k )  (k  1) (k  1)  1) 
k 2


 
2 .2898681336964528729...
1
= 1   2

2
k 1 k ( k  1)

2
3

=

0
1 

 k  (k  1)   (k  2)  2 
k 1
x
1 e
dx 
ex  1
GR 3.411.25
1 x
log x dx
1 x
0
1
=
3 .2898681336964528729...


2
3


 2(2) 
( k  1)( k )

2 k 2
k 3


 
log 2 x dx
x 2 dx
= 


sinh 2 x 0 (1  x ) 2

 
log 2 (1  x )
log(1  x )
= 
dx   
dx 
2
x
x
0
0




1

GR 4.231.4
 
x 2 dx
=  x

e  e x  2
0
1
log 2 x dx
1 ( x  1)2 
GR 4.231.4


 
=
e
dx

1
6 1
.28989794855663561964... 

5
 1
.2899379892228521445...   1   (1  log( e  1)) 
 e
x1 / 3
0
57 .289961630759424687...

cot 1  cot

CFG C1
( 1) k 1 Hk


k
k  1 ( e  1)


180

29
1
.290000000000000000000 =
 2
100 k 1 k  7k  6
1 .29045464908758548549... 
(1) k 1 H O k

2k
k 1

2
1
arctan

3
2
.29013991712236773249... 
21/ e 

2
( 1) k / k !

k 0

.29051419476144759919... 
sin x dx

ex  2

0

 
1 
1
1


,  i ,    2 csch ( ) cos(log(2 )) 
 2 F1  i ,1,1  i ,    2 F1   i ,11


4 
2
2
=

2k
 (k  1)  1 

k 1 k  2

1
2 .29069825230323823095...  e erf (1)  

1
k  0 ( k  2 )!
.29053073339766362911... 
1
3  log 2  
4

 3

1
1

  2
2
6
4 3
k 1 4k  3

1
( 1) k
.29081912799355107029...  4  8 arctan   k
2 k  0 4 ( 2 k  3)
1
1
log(1  ( e  1) x )
.29098835343466321219... 
 
dx 
2e  2
1  ( e  1) x
0
.2907302564411782374...

.291120263232506253...

2
24
coth



log 2 2

4
2
k 1
1
k 1
k2
log(1  x 2 ) dx
0 x(1  x 2 ) 

GR 4.295.17
1
=
GR 4.295.17
log(1  x 2 / 2)
dx 
x
0
1
=


.29112157403119441224... 
 5k
k 1
1

3
1

 ( 3k )
k 1
5k


2 k 1 k

(2 k  1)!( k  1)
k 1


1
1
dx
 2 coth     2   2
 2

4
x 1
k 1 k  1
1

.2912006131960343334...

2  2 cos 2  2 sin 2   ( 1)k 1
.2912758840711328645...

 1/ ( k
1 .291281950124925073115... 
3
24
2
 1)


1
1 .29128599706266354041...   k 
k 1 k
1
dx
x
x

GR 3.486
0

5 .2915026221291811810...


.29156090403081878014... 
.29162205063154922281... 

 .29166666666666666666
=
28  2 7 
G


1  2  2 log 2 7 (3)




4 16
4
8
7

24

.29171761150604061565... 

k 1
.29183340144492820645... 
=
.2918559274883350395...

.2919265817264288065...

=
1 .29192819501249250731... 


( 1)

k
2  2)
 k(
k 1
(2 k )
3k
1

0
x 3 arccos2 x
dx 
1  x2
dx
 ( x  1)
4
0

635
 31 / 4

cot  31 / 4  coth  31 / 4 
1107
4


3

3k ( ( 4 k )  1)   4

k 1
k 2 k  3


 (k )
  1 1

 Li3     


3
 k k
k 2 k
k 1 

2 2 k 1
cos2 1  1   ( 1) k 1

( 2 k )!
k 1

( 1) k 1 4 k k 2

k 1 ( 2 k )!( k  1)

3
arctan 2 x dx

 
24
1  x2
1

.291935813178557193...
k 1
1
 ( 1)
k 2
k
 (k )  1
k!
1



  e
k 2
 1/ k
1
1  
k
.292017202911390492473... 
3 G

   x arccot x log x dx 
4 2
0
.29215558535053869628... 
( 1)k


k  2 k !(( k )  1)

GR 1.412.2
.29215635618824943767... 
1
2
2


6

coth  3/ 2

2 

k
k 1
1
2
(k 2  )


.29226465556771149906... 
 (2k  1) (2 k  1)  1 
k 1
.292453634344560827916... 

 
.2924930746750417626...

=

.29251457658160771495... 
.29265193313390600105... 
.2928932188134524756...

=

1
3    log 2     (k )  1 
2
k 1
k 2

1
1


   k log 1    1   

k
2k 
k 2 

1
1



coth

  2
2 5
5 2
k 1 5 k  1

sin 2 k cos2 k
1
cos 4

e  e cos(sin 4)  
k!
8
k 1

H ( 3) k


k
k 1 4 k
1
1

2


 2k 
(2 k )!
k 1 1
(

1
)
( 1) k 1






k
4 k
( k !) 2 4 k
k 1
k 1


 .2928968253968253
=
H
7381
 10 
25200
10

1


2
k 1 k  10 k

k
k 6

2 .292998145057285615802... 
3 .293143919512913722...
 3020 .2932277767920675142...



2 PFQ  1,1,1, 2, 2, 2, 1   2
2
1
 25
1

2
k 1 k!k
1 1 3 
 ,  ,1 
2 2 2 

1
e
x
log 2 x dx
0
3/ 2
k

k
k 1 2

7 
log(1  x )
dx 
1  x3
0
1
.29336724568719276586... 

.29380029841095036422... 
sinh 1

4
1 .2939358818836499924...

.293947018268435532717... 

 1  dx
4 5 
x
 cosh x
1

 (k ) log (k ) 
k 2



 
=
1

48 (G  1)   3  96 log 2  63 (3)  6 2 2  7 log 2 
192

arctan 3 x
1 x 4 dx

 .2941176470588235
.2941592653589935936...
=

5

17

coth 5 
1

50

k
2
1

 25
10
k 1

i (1)
2
.29423354275931886558... 
 (2  i )  (1) (2  i )   3

2 2
)
2
k  2 k (1  k

J124



 
=
6 (1) k 1 k  (2k  1)  1
k 1


 
=
xsin x
x
1
 e e

x
0
.29423686092294615057... 
( 1) k  ( k )  1


k2  k
k 2
.29430074446347607915... 
(1) k


k
k 0 2  1

1 1
 k 1 
log 1     

k
k k
k 2

 

.2943594734442134266...

I2 ( 2 ) 
 k !(k  1)!2
k 0
3
=

0
.2945800187144293609...

(sin x  x cos x)3
dx
x5
4  

.29474387140453689637... 
1
  4 
k 2
2
12
 2 log 2 
 
=
1


 ( k )
2 .294971003328297232258... 
2e   
k 2
k3
4 .29507862687843037097...   k
k 1 k

(k )
 k ( k  1)
k 2
2 .2948565916733137942...
(1) k 1


3
2
k 1 2k  k



( (k )  1) 2
1 
1 



1


 (k )
 (k ) 
k 2
k 2 


J1061
Prud. 2.5.29.24
 ( k  2)  1

 (k )  1 

.29478885989463223571... 





k

k6  k

k 2 ( ( 3k )  1) 


3
3
k  2 ( k  1)
k 1

3
k (k  3)
 
 .29452431127404311611... 

3 2
32
k 1 ( k  2 )

.2943720976723057236...

k

.29522056775540668012... 
1 .295370965952553590595... 

 .2954089751509193379...

.29543145370663020628... 


I 0 (1)  L0 (1)  
2
1

2e
e
x
arcsin x dx 
0
G 2

6
2 log 2
1 
3

 2k
k 1
2
1

 5k  2


 
=
log(1  x)
dx 
x4
1

1


=
x
2
0
3 .295497493360578095...
 .2955013479145809011...
 .2955231941257974048...
57 .2955779130823208768...




e  
1
2
   (1   )   k ( k 1 
k 1


180


k 1
k (k  )
2
)

H 2
2


, the number of degrees in one radian
 ( k )( 1) k
4k
k 1
7 .29571465265948923842... 

2
 (1  )     

.29559205186903602932... 
 1
log1  dx 
x


  3 csc  3 
k 2  2k

2
k 1 k  2 k  2

(1) k 1


2
k 1 4 k  4 k  5

.295775076062376274178... 

  x 1  e 3 x  dx
0  3e  x  x 
1 n 1
x  x n  2 / 3  x n 1 / 3  3 x 3n 1
3 .29583686600432907419...  3 log 3  
dx 
1 x
0
.29583686600432907419... 
.29588784522204717291... 
.29593742160765668...
5 .2959766377607603139...
.29629600000000000000
3 log 3  3 
 log 2 2
2

 2
12

 2 log 2
12
  (3) 
k2


k
k 1 2 ( k  1)( k  2 )
cosh   1
 e   e   2

 sinh 2 

2
2
4
37037
=
, mil/mile
125000
 10  14 log 2 

 ( k  1)
k 1
2k k 2



 4k
 (4k )!
k 0
GR 3.437
GR 3.272.2
1 
9  3

  x 3 log1  3 dx 
12
 x 
0
1
.2965501589414455374...

.29667513474359103467... 

 

 .29697406928362356143... 

 .29699707514508096216... 

 .297015540604756039...
=

.2970968449824711858...

1 .2974425414002562937...

3 .2974425414002562937...

.29756636056624138494... 
.2975868359824348472...

.29765983718974973707... 
 3  5  1  5  2 
 

  1 

 2 
2
2






1 

1  2


k k
k 1 

4
k
7 ( 3) 


12 k 1 ( k  12 ) 4

1
3
2k k
k 1


  ( 1)
( k  1)!
2 2e 2 k 0

1  3
 3
1

tan
 2

2 12
2
k 1 4 k  4 k  2
3   (1) k 1 (2k )!!k

22 F1  2,2, ,1  
2  k 1 (2k  1)!!


1
2 e  2 
k
k  0 ( k  1)! 2


2k  1
pf ( k )

2 e  


k
2k
k 0 k !2
k 0

(1) k 1
1 
 


6 24
k  0 12 k  6

1
cos
 5
1  5

 3 log 2 
6
 2 
2
1  4
  
3  3


( 1) k ( k !) 2

2 
k  0 ( 2 k )!( 2 k  1)

Berndt 32.3
x 3 dx
0
ex
3





 (1) k 1 
1 .29777654593982225680...   1 

 2k  
k 1

  

k  


.2978447548942876738...

Hk 3


k
k 1 6

  arctan x dx
1
.29790266899808726126... 
   2  2 log(3  2 2 ) 
4

H (2) k
.2979053513880541819...   k
k 1 4 k
e
.29817368116159703717...   Ei ( 1) 
2

xe  x dx
0 x 2  1 
2
1
2
0
  3 (k )


 1 

k  2   (2k )


k 2
1
.29827840441917252967...   2 log

k
k 2 k
1
1  log k
.2984308781230878565... 
 (2)   2

 k 1 k

4 .298268799186952004814... 


3 .298462247044795629253... 

2
log(3  2 ) 
( 1) k


k
k 2 k  1

.29849353784930260079... 



3 .2985089027387068694...

log( x 2  9)
0 x 2  2 dx
1
( 1) k

 jk 
k  2 j 1 k


32  4
, volume of the unit sphere in R9 
945
log 6

6

e2
5
2k
.29863201236633127840... 

 
8
8 k  0 (k  3)!
.2986265782046758335...
.2986798531646551388...


 
 log 3 

3 3
3 3
1
=

 


0
x dx
3
(e 3 x  1) 2

x log x dx
(1  x 3 ) 2

k  1  (2 k )  1
.29868312621868806528...   ( 1)

k 1
k 1
15
3 log 3 
1
.2987954712201816344... 
 24  3 

16
8
k 1 k (3k  4 )

8
8
1
1
1 .29895306805743878110... 
arcsin

 

7 7 7
2 2
k  0  2k  k
 2
k 
0
3

.29933228862677768482... 
2
4

1 .2993615699382227606...

(1) k k (k  1)


4k
k 1


2


1
.29942803519306324920...   ( 2 ) 

cot

2 2 2
2


( k )
k
k 2

 2G  4 
 

=

k 1
2
( 2 k )   ( 2 k  2 )
2k

k2 1


4
2
k 1 2 k  k

GR 3.456.2
GR 4.244.3

( 1)k


3
k 2 k  5
6
.2994647090064681986... 

3
e  e 3

1 e
cosh k
1/ e
cosh 1
8 .29946505124451516171... 
e  e  e
cosh(sinh 1)  


2
k!
k 0
.29944356942685078...
2 .2998054391128603133...



  3  1 


 log1  x
0
.299875433839200776952... 

2 
 dx 
 1
G 1  log 2

 

48 3 3
4
2
2
5 .29991625085634987194... 
2

arctan 2 x
1 x 4 dx 
3
1 1
3 1
1
 2
  
 ( , )     1   
3 3
2  3 
 3
 3
J132
GR 1.471.2

.300000000000000000000 =

=
1 .30031286185924454169... 
1

cos
2

.30051422578989857135... 
 (1) e
k  ( 2 k  1) log 3
J943
k 0
cos x
dx
e3 x

0
 1  4i
2
 (k ) 0 (k )
k 1
.300462606288665774427... 

sin 3 x
0 e x dx 

.300428331753546243129... 

3
1


10
2 cosh log 3
2k  1
cos
 1  4i
2



 1  k
k 1

2

1

2 
(k  1) 

13
12
2
 (3)
 1
 1
 2 (3)  log 3 2  2 Li2   log 2  2 Li3  
 2
 2
3
4
[Ramanujan] Berndt Ch. 9

=
log 2 x
1 x 3  x dx 

log 2 x
1 ( x  1) x( x  1) dx
2
=
log 2 x
1 x  1 dx
[Ramanujan] Berndt Ch. 9
1
=
x log 2 x
0 1  x 2 dx

=
x 2 dx
0 e2 x  1
=
log 2 (1  x)
0 x dx
=
log 2 (1  x )
0 x dx 
1


1
log(1  x 2 ) log x
dx
0
x
1
 /2
=
 log sin x
2
tan x dx 
0
2 .300674528725270487137... 
2 ( 3) 

.300827687346536407216... 
2
k 1
1 .3010141145324885742...

.301029995663981195214... 
k
Hk

(2 k  1)
(3)( 4) 
log10 2

 0 (k )
42 .30104750373350806686...  
k!
k 1
 4(3)
90


k 1
1 ( k )
k
4


k 1
 1 ( k )
k3

HW Thm. 290
.3010931726...


p k prime
.30116867893975678925... 

1

pk pk 1
sin 1  cos1 
=
(1) k 1


k 1 ( 2k  1)!( 2k  1)
=
 1  dx
0 x sin x dx  1 sin x  x3


 
(1) k 1 (2k  1)


(2k )!
k 1


1
e
=
1 .301225428913118242223... 

 
=
1 .30129028456857300855... 
log x sin log x
dx
x
1
90
2
10800 (4)    6 (2)  60 4 (4)  360 2 (2) (4) 
10





 (k ) log 2 k
k 1
k2

1  x 4
dx 
2  1   log
1  x2
0



=
 (x
2
0
2 .3012989023072948735...

 



 log1  x
0
2
1 
 dx 
 1
dx

 1 / 2)( x 2  1)


 Re{sin(log i )}  i cos (1  i )  
2
2



1

=   1 
2 
(2 k  1) 
k 0 

.301376553376076650855... 
sinh

6


2

9
1
x
2
arcsin x dx
0

sin 2
2k k 2
  (1) k  1
( 2 k )!( k  1)
2
k 1


1
 (k )
  0k
.301733853597972457948...   k
5
k 1 5  1
k 1
.301544001363391640157... 
1
.301737240203145494461... 
log 2 3

4
1 .301760336046015099876... 
1 .3018463986037126778...

 
log(1  2 x )
dx
1
2
x

0
1

2 arctan e 2 

2
 arcsin tanh 2  gd 2


e  e  
 (2 k )
1

  ( 1) k 1
  log 1  2  

k
k 
2
k 1
k 1
 sinh  
= log
   log  1  i   log  1  i 
  

log
GR 4.523.1

4k
11 .30192195213633049636...  I 0 ( 4 )  
2
k  0 ( k !)

 2 k  1 k
40
.301996410804989806545...  8 
 

k  16k
3 3
k 1 
.302291222970795777178... 
.30229989403903630843... 
cos( 2 k  1)
2k  1
k 1
k

(1)



6 3 k  0 (3k  1)(3k  3)
1
1
log cot 
2
2


 
=
dx
0 x 3  8 

=
3
x dx
1 x
12
0



x dx
1 x 4  x 2  1 

GR 1.442.2

 (x

2
dx

 3) 2
7
x dx
12
0 1 x

H k  (k )  1

2k
k 2

.30240035386897389123... 

2 .302585092994045684018... 
5 .30263321633763963143... 
=

log10

1
 3
56 (3)  2 3    ( 2 )    2
3 3
 4
k  0 (k  4 )

(k  1)(k  2) (k )


4 k 3
k 3


.30276089444130022385... 
  (2 x)  1 dx
1
1 .30292004734231464290... 
.303150275147523568676... 
.303265329856316711802... 
3 .30326599919412410519... 
.303301072125612324873... 
=
2
12
1
log( x / 2 )
dx
1 x
0
 log 2   
2
 2
log   
 3
(1) k 1
 
k
2 e
k 1 k ! 2



 2   3
csc sec       
 5   5
2
5
5
1

 cos1  
 cos1 
 sin 1  
1  cos 
  
  cosh 
  sinh 
 2 
 2 
 2 
 sin 1
cos


 2 
cos k
1

(1) k 1

(cos 1) / 2
e
k !2k
k 1

1 .303324170670084184712... 
 H  (k )  1
k 2
.303394881754352755635... 
( 2)
k


 1  x 2  x 2 dx
(log 4  1)   log 2 
4
 x  (1  x 2 ) 2
0

.30342106428050762292... 
1
 (k  1)k (k  1) log k
k 2

1 .303497944016070234172... 
GR 4.298.19

1
 log 1  k !
k 1

8  2 log 2
(1) k 1
.30378945806558801568... 
 
 
9 3
3
k 1 k (k  3 / 2)

5


 (2 k )  1
.303826933784633913104... 
coth
   ( 1) k 1

2k
2 2
2 6
k 1

1
= 
2
k  2 2k  1
  1
 1 
.30389379330748584659...  2 Li3     Li3     
 2
  3
.3039485149055946998...

.30396355092701331433... 
.30410500345454707706... 
.304186489039561045624... 
.304349609021883684177... 

 
=
2 .304632878711566539649... 
2 .304988258242580902703... 
2
12
3

14

27
2
1
log 2 x
0 ( x  2)( x  3) dx
1
x
2
arcsin 2 x dx 
0
( k  12 )!
3 2 2   
k
k  1 ( k  1)! 2

1

log 2

 
2
24
4
k  1 16k  12 k
1
1
sinh 
  log (2  i ) (2  i ) 
log
2
2
2


k

(
2
)
1
( 1) k 1


2k
k 1


1


csc  2   1  2


k  2k  1 
2
k 1 

3 
(1) k 1 k 2

  2
3
8
2
k 1 ( k  1 / 4 )



3
.305232894324563360615... 
.3053218647257396717...

log 2  2 log 2  2 
G
3
 log  (x ) dx
2
GR 6.441.1
.305490036930133642027... 
.305514069416909279097... 

2  log 2 2 
k
 k !( k  7) 
5040  1854e 
k 1
3
2 log 2  2
2 log 2  1  2 log 2   (3) 

3
6


H ( 2) k
 
=  k
k 1 2 ( k  2)

1
.305548232301482856123...  

k  3 k ! 2
3 2
1
.30562962705065479621... 
arcsin
1 
2
3

1
.305808077190268473430...   3
2
k  2 k log k
.305986696230598506245... 
.306018059984358556256... 
138

451
6

.306119389040098524366... 

1 .30619332493997485204... 
13 .30625662804500320717... 
.306354175616294411282... 
1 .306562964876376527857... 
=
.306563211820816809902... 
8 .3066238629180748526...

.306634521423027913862... 
3
( 2 k )!!
 (2 k  1)!!3 (2 k  1)
k 1
k
3
Fk

k
k 1 6
1
 2 ( 3)    log(1  x ) log 2 x dx
0
 (k  2)  1
k 1
.30613305077076348915... 
2



e1/ k
  3
k 2 k
( k  1)!

4 3 1
 
27
2
 (x
2
0
dx
 x  1) 3

3 sin 2 3 cos 2
3 
( 1) k 4 k k 3


J 3/ 2 ( 2 )  
4
2
2
( 2 k )!
k 1
k
k

128
( 1) (3  1)( k  1)
G 
 ( k  2)
9
4k
k 1
1  3
  
4  4

x e
2 x4
dx
0
2 2


 cos  sin 
2
8
8
k


( 1) 
1 


4k  2
k 0 
 2 
1  J0   
 3
(1) k  1

2 k
k  1 ( k !) 3

69
1 
 1 
 79 4  15 ( 3)    
1215 
 3 
2 sin
5
8

=
log 3 x
dx
x3  x2  x

1
.30683697542290869392... 
.306852819440054690583... 

4
coth  
2
1  log 2 
4
csch 2  
( 1)
k
k 2


k

1

2

 (k
k 1

 ( 1)
k
k 2
2
1
 1) 2
 (k )
2k

k 1
k
k
k 2

1

k
k  1 2 k ( k  1)


1
(1) k 1
2



3
k 1 k ( 4 k  2 )
k 1 (2k )  2k

Hk

k 1 ( 2k  1)( 2k  3)

=
=
=
=
2

=


 
=

dx
1 x 3  x 2 
1


2
k 1 k  k

x
1
J145
J149
dx
x
2
dx

x
 1
 e e
x
0

=
 x
e2 x  dx
 2x
0  e  e  x  x

=
  (x  1)e
x
 e x / 2 
0
1
=
dx
x
x  1  dx

  x  log x  log x
GR 3.438.3
GR 3.435
GR 4.283.1
0

=
xe x

e2 x  1
0

=
x
dx
log x
2
1
x2  1
GR 3.452.4
dx
GR 4.241.8
 /2
=

 log(sin x ) sin x dx
GR 4.384.5
0
1
1 .306852819440054690583... 
2  log 2    arccos x log x dx
0
.30747679967138350672... 
3 arcsinh 1  2

4
 2k 
( 1) k
 

k
k  0 4 ( 2 k  1)( 2 k  3)  k 

GR 4.591.2
1

 
=
 x arcsinh x dx 
0

3 ( 3)  2
1
(1) k 1


  3
8
48 16
k 1 k ( k  2 )
4
.307692307692307692307692=
13

(1) k 1
 1/ e
.307799372444653646135...  1  e
 
k
k 1 k ! e
.307654580328819552466... 
 log(cos 1) 
sin 2 k

  ( 1) k 1
2
k
k 1

(1) k
.308169071115984935787...  arccot    2 k  1
( 2 k  1)
k 1 
.30781323519300713107... 
J523

16k
e 4  e 4
 
2
k  0 ( 2 k )!

k 1  ( k )
.308337097266942899991...   ( 1)

3k  1
k 1


2
1
( k  1) ( k )
.308425137534042456839... 
 



2
32
2k  2
k  0 (4k  2)
k 2
27 .308232836016486629202... 
cosh 4 
AS 4.5.63

1

 
=
.30850832255367103953... 
1 .3085180169126677982...

.30860900855623185640... 
x log x
0 x 4  1 dx 

x log x
1 x 4  1 
1
arctan x
1  x2
0


I 0 (2)  (1)1 (2k )!
(1) k

 
e2
k!
(k!)3
k 0
k 0
3
2
1 1 1
 6 log 2 
4
0 0 0

(1)
k 0
22

4
5

1 .308996938995747182693... 
12

 2k 
 
k 
x yz
dx dy dz
1  xyz
k
k
2 .308862659606131442426... 

 48k
k 1
(1) k / 3
= 
k  0 2k  1
 1
.309033126487808472317...   Li2    
 3
2
1

 48k  9

Prud. 5.1.4.5
=
1 2
 1 
 log 3  2 log 3 log 4  4 log 2 2  2 Li2   
 4 
2
=
1 2
 3 
   3 log 2 3  3 log2 4  6 Li2   
 4
6
=
(1) k  1


k 2
k 1 3 k


Hk
k
k
4
k 1
1
=
log x
dx
0 x3

 3
2
9 log 3


9 
2
6
2

k 1  (k  2)
=  ( 1)
3k
k 1
4
4 .309401076758503058037...  2 
3
1
2 .30967037950216633419...   
.30938841220604682364... 

 3k
k 1
3
1

 k2
CFG A10


2 .30967883660676806428... 
16 .309690970754271412162... 
.309859339101112430184... 
3 log 2
log( x 2  2)
 
x2  2
2 2
0
6e



1
1

  2

cot
2
2 6
6
k 1 6k  1

 ( 2k )
k 1
6k

.30989965660362137025... 
 (i )  (i )
.309970597375750274693... 
log 2 1   1  i 

 1  i
  
 
   (i )   (i ) 
 2 

2
8  2 

=
sin 2 x
0 ex  1 dx

.310016091825079303073... 
 ( 1)
k 1
 (2 k )  1
k ( k  1)
k 1



  ( k
k 2
2
1


 1) log 1  2   1


k 
1
1
1
 (2  log2  2 cos1) sin  cos  
2
2
2

k
1 .310485335489191657337...   k
k 1 2 Hk
.31034129822300065748... 
1 .310509699125215882302... 
2
3


8
coth  


 
=
3
3 2
csch 2 
coth  csch 2  2 ( 3) 
8
4
 (1) k  (k )   (2k )
k
k 2
2
(1) k 1
2k  1
sin

2
k 1 k  1

6
7


9
k 1 k
k 1 k
3 .310914970542980894360... 
  (1)
  (1)
e
k!
k!
k 1
k 1

159

1
.311001742407005668121...  

cot  14   2
1820 2 14
k  1 k  8k  2
1 .3110287771460599052...


 

1
1
 2   , the lemniscate constant
4 2  4 
=
1  3 2  3 
   
2 2
 4
5
 4

1
2 .311350336852182164229...  
k  0 k ! k !!
 
 3
 4
1

    1   
=
 /2
2  1 
K
 
2  2
0

1  sin 2 x dx 
0
dx
1  x4

2 .31145469958184343582... 
2
1 

   log x log 1  2 2  dx
 e x 
e
0



 (1)k 1
 ( 2k )
k 1
4k
1
.311612620070115256697... 
log 1  2 
2
arcsinh1 
4
2 2

 sinh    
1
.311696880108669610301...  
csch 2   sinh    2  

2
8cosh   1
 16

 
=
.31174142278816155776... 


4k 2


2
2
k 1 ( 4k  1)
1
 sin 1 
 cos1
 cos1 
 sin 1
sin
  cosh
  sinh
   (cos 1) / 2 sin

 2 
 2 
 2 
 2 
e
( 1) k 1 sin k
= 

k !2 k
k 1
1

 (1  e)  Ei (1)  e Ei (1)  
e

.311770492301365488...
1 i 3
2i

Li2 
3  3i
 2 
.311821131864326983238... 

 (1)
k 1
k 1
Hk
(k  1)!
1 i 3
3 i

Li2 
3  3i
 2 
5 2 1 (1)  1
x log x
     
dx
 3
36
6
1  x  x2
0
1
=

=
 e (e
x
0
.31182733772005388216... 
8 .31187288206608164149... 

7
x
x
dx
 e x  1)
tanh
 7
 7
2

7

8
GR 4.233.4
GR 3.418.2

k
k 1
2
1
 5k  8
.311993314369948002...

 
1 .312371838687628161

log 2 2
3 log 2
 1 

  log 2 

2
2
2

( 1) k k  ( k  1)
= 
k 1
k 1
1920 
1 
1 
1
=
 1   1   1   
1463 
7   11  19 
1 
1 
1 
1 

2 1  2   1  2   1  2   1  2  

3 
7   11   19 
=

 


[Ramanujan] Berndt Ch. 22
3 ( 3) 

 2  4 log 2  8 
4
6
(1) k  1
.312382639940836992172... 

3
k  1 k ( 2 k  1)

2k
2 .3124329444966854611...  2 HypPFQ{1,1,1,1},{2,2,2,2,},2  
3
k 1 k!k



5
k
1
1
.312500000000000000000 =
  k   3


1
2 2
3
)
16
k 1 5
k  2 k  2k  k
k  2 k (1  k
2


=
 k  (2k  1)  1
k 1

2 .3126789504275163185...


 (2 k )
2


k
k 1
k 1

( 1)
.312769582219941460697...  
k  1 k ! ( k  1)
k 1
.312821376456508281708... 
2

e3
 1
2

 Li  k
2

cos 3x
dx 
(1  x 2 ) 2



i  
i 
 ( 2k )  1

 log  2 

 2 
   (1) k 1
2k k
2 
2

k 1

 1 log 2
arctan x
.312941524372491884969... 
dx
 
 
x4
12 6
6
1
.31284118498645851249... 


2
2
Bk 2 k
.313035285499331303636...  (coth 1)  1  2
  2k  
e 1
k 1 e
k 0 k !
2
k

4 B2 k
e 1
1 .313035285499331303636...  coth 1  2
 i cot i  
e 1
k  0 ( 2 k )!
2
k

2e
B 2
  ( 1) k 1 k
2 .313035285499331303636... 
2
e 1
k!
k 0
.313165880450868375872... 
arccsch 
AS 4.5.67
.31321412527066138178... 

 

log 3 
4

 arctan (3k
k 1
k
3
1
 1 
   1    

e
e 

e

log(1  e)
.313298542511547496908... 
 ( 3)  1

 (2)  1
 
.3133285343288750628...
=

 
.3134363430819095293...

32
=
k 1
ek

( 1) k 1

ek k
k 1


1 
(1) k 1 (k  12 )!


4 2 k 1
(k  1)!
2 log 2 
2
x
 3
3
log 3 

2
2

 6k
k 1
1

k
2
log(1  x )
dx
x2
0

6

1 
i 
i 
1

    1    1      3

2 
3
3
k  1 3k  k

 (2k  1)
1  i 
 i 
=    

   
    (1) k 1
2  3
3k
3 

k 1
 (2)
 .313666127079253925969...  log
 ( 3)
.313535532949589876285... 


xe  2 tan
1
 
 (k )

2  cos 2 x
dx
0
cos 4 xcot x


23
1
1
 2e  



4
k 1 k !( k  4 )
k 1 ( k  1)!  3k !
=
.313513747770728380036... 


1 1
2
1

I (2)  2 I1 (2) 
1 F1  , 3,  4  
2 0
2 2
3e
2e


1  2k 
 
(1) k

(k  2)! k 
k 0
 /2

1


2 2
k  1 k e  ke
dx
1
1 .313261687518222834049... 
.31330685966024952260... 

x
1

10k

 2 )( 9 k 2  1)
1
5
 1
.313261687518222834049...  log 1   

e
=
2
[Ramanujan] Berndt Ch. 2
k 1
.313248314656602053118... 



.313232103973115614670... 

GR 3.964.2
(1) k 1
.313707199711785966821...  
k
k  1 k ! 2  ( 2 k  1)

30 .3137990863619991972...

Titchmarsh 14.32.3
1
1 
1
  16, 3,  
2 
4


0
log 2 x
dx
x 4  1 / 16
3
 2
 1
 hg    hg   
 3
 3
3 2
4


H ( k  1)  1
1
2 .31381006997532558873... 
 2 log 2   k
 2  3
k 
72
2k  1  k
k 1
2

  ( 2 ) 2  ( 2 )
 ( k ) log k
.3139018813067524238... 

 3
 
2
 (2)  (2)
k2
k 1
.313799364234217850594... 
1 .31413511119186663685... 
.314159265358979323846... 

3 2  4


4
16


x3
0 sinh3 x dx
10

7129
H9
1
.314329805996472663139... 

  2
22680
9
k 1 k  9k
2
1036 
1 .3145763107479915715... 

 H ( 2 ) 5/ 2
225
3
( 3)

H k
.31461326649400975195...   k
k  1 2 k ( k  1)

 ( k  1)
2 .31462919078223930970...  e  e Ei ( 1)  1  
k  1 ( k  1)!
(1) k 1 H ( 3) k
.31506522977259791641...  
2k
k 1

( 1) k 1
1
1
.315121018556805747334... 
sin
  k
3
3
k  1 3 ( 2 k  1)!


 3
2  2

1
1 .31521355573534521931...  
k
k 1 (1  5 )
2 .31515737339411700043... 
.315236751687193398061... 
2 .315239207248862830396... 

e
0
x dx
1
x
e2
3


log 2 x
dx
 log 2   2
x 4
16 4
0

.315275214363904697922... 
    i (1 / 2) 1 
 (1)
k 1
2
k 1
k
 ( 3k )
2k

2k 3
 
3
2
k  1 ( 2 k  1)

.31546487678572871855... 
log9 2 
3
1
.315496761830453469039... 

.31559503344046037327... 
.31571845205389007685... 
 
dx
 3 x
3  3i  6  37 / 6 i  32 / 3 
3  3i  6  37 / 6 i  32 / 3 




6  35/ 6 
6
6
 6  35/ 6 

3
1 

5/ 6   1  1 / 3  

3 3
3 


 (3k )
1
  (1) k 1 k

3
3
k 1 3k  1
k 1

 
=

x
=
9 e 15 
erfi 1 

8
16

 (k )
k 2
k


 p (k )
k 2
k


k
 k !(2k  5)
k 1
log (k )   
 
1 1
 log 1     

p  p 
p prime 


6

19
1
tan x dx
.3158473598363041129...  
ex
0
.315789473684210526
=
1
csc  2 sin  3 
6
.315888571364487820746... 


 1  k
k 3
2
1


 2k  1 
1
.3160073586544089317...

 log(1  log(1  x)) dx
0
.316060279414278839202... 
1
1


2 2e

1
 xe
x2
dx 
0

 1  dx 1
 1  dx
1 cosh x 2  x5  2 1 cosh x  x3
1/ 4
1 .316074012952492460819...  3
=
10
1
 sin arctan 
10
3
log 2
log 3
.3162417889176087212...  log 2 (log 3 
  )  log 3(  
)
2
2

k  (k )
 
=  ( 1)
2k k
k 1
1

log( k  1)
 3  x  dx
.316281418603112960885...    log
 

 2  log x
3k k
k 1
0
.3162277660168379332...


GR 4.221.3

Hk

(k  1) 2
k 1



 ( 2k )
 (4k  2)
1
.31642150902189314370...   k    k



4k  2
2
1
k 1 2
k 1 4  1
k 1 4

k2
1
.3165164694380020839... 
I 0 (1)  
2 k
4
k  1 ( k !) 4
.316302575782329983254... 
3 (3)  2 (2) 
k
2
1
6 .31656383902767884872... 
Ei ( e)  Ei (1) 
e
ex
dx
0
3 .316624790355399849115... 
11 

 2k  1
2 (2 log 2  log(2  2 )     k 
k 1  k  8 k

( 2 k  1)!!
= 
k
k 1 ( 2 k )! 2 k
.316694367640749877787... 

 
.316737643877378685674... 
.316792763484165509320... 
1 .316957896924816708625... 

 
.3171338092998413134...
=

5 .317361552716548081895... 
1
1
 3 
3 3e
e
dx
x
4
1

cosh 1  3 sinh 1
e
1
k4
 
 
16
8 16e
k  1 ( 2 k  1)!
1
arccosh 2  2 arcsinh
  log 2  3 
2
3 1
log
3 1

( 1) k  ( k )
1    2(   1) log 2  log 2 2  
k 1 k ( k  1)
3 



.317418282448647644165... 
13 (3) 

2 3
3 3

28

8

1
 (k  2 / 3)
k 1
3
(1) k 1
.317430549669142228460...  1 
 2 2
2 sinh  / 2
k 1 4k  1

log 2 x
13  2 11
1 .31745176555867314275... 


log 2  
dx
( x  1)5
48 24 12
0
3
3
3 / 2
111 .317778489856226026841...  e
 cosh
 sinh
 i  3i
2
2


7 4
log 3 x
x3
.317803023016524494541...  6 
dx
  3

0 ex (ex  1) dx
120
x  x2
1
1
.31783724519578224472... 
3 2
1 .3179021514544038949...

1


k 1 k ! k

eEi(1)   


 
k Hk

 (k  1)! 
k 1
e
1
1
= 
   log log x dx    e x log x dx
(
k

)!

k
!
1
k 1
1
0
1 
1
1 

 2 F1   i ,1,1  i ,   2 F1  i ,1,1  i ,   

2 
2
2 
3 log 3

 2
1 .318234415786588472402...        

 3
2
2 3
1 .318057480653625305231... 
.318309886183790671538... 
1



2
k 0
k
1
( k 2  1)
GR 8.366.7
2 2  (4k )!(26390k  1103)
Ramanujan, Borwein-Devlin p. 74

9801 k 0
(k!) 4 396 4 k
(1) k (6k )!(545140134k  13591409)
Borwein-Devlin p. 74
(3k )!(k!) 3 640530 (3k 3) / 2
k 0

k (k   )
J1061

k 1 ( k    1)( k  1)


 


12

1

 
=
 x sin  x dx
0
.318335218955105656695... 
=
1
 coth   3 2 csch 2  2 3 coth  csch 2  

8
1
  i (1) (1  i )  i (1) (i  i )   ( 2 ) (1  i )   ( 2 ) (1  i )
8
2

k  k k 2  1
 k
k 2

=
 ( 1)

1
2
k 2  (2 k )   (2 k  1)
k 1
k 1
2 .318390655104304125550... 
cosh 

5
3324
5 .318400000000000000000 =

625
(1)  7 
1 .31851309234965891718...    
 6
37 .31851309234965891718... 
.318876248690727246329... 
3


4
 1  (2k  1)
k 1
2




Fk 2 k 2

k
k 1 4
 1
 6
 (1)  
3

4 e2
 /2
 cos(2 tan x ) cos
2
0
x dx
GR 3.716.5
3 5 
  3  5 
2 
  Li2 
 
arccsch 4  2 log 5  Li2 



2
2
5 5 





k 1 Hk
=  ( 1)

 2k 
k 1
 
 k
.318904117184602840105... 
 
 (2k )  1

.319060885420898132443... 
 (1)k 1

( 2 k )!
14
.319135254455177440486...  4 2 arcsinh1 

3
k 1
1 .319507910772894259374... 
.319737807484861943514... 

1
 1  cos k 

k 2

1
 2 (2k  5)
k
k 0
1/ 5
4 
 (k  1)

 log 2  log 2 2   
2 (k  1)
k
k 1

 2 (1) k
k 2

k  1 sin k
.31987291043476806811...  arccot  cot 1  2 csc 1   ( 1)

2k k
k 1

log k
.32019863262852587630...   3
2
k 2 k  k
.32030200927880094978... 
2 log 2 2

3
.320341142512793836273... 
 (1)

k 2

.32042269407388092804... 
1
k
log(1  3x )
dx
1  3x
0
1

 (k )
k
2



1
1
2
k 1

 (n  1)!  (1)
n 1


  Li   k   k 
k 1
k 2
log n k
k!
k
9 .320451712281870406689...  
k  1 Fk
(1) k 1

k 3
k 1 3 k


 (2k )  1
1
1
k 1
.320756476162566431408...   ( 1)

sin

2 k 1
( 2 k  1)! 2
2k
k 1
k 2 k

 1
i
sin k / 2
1 .320796326794896619231... 
  log  ei / 2   

k
2 4
2
k 1

1  ei / 2
sin k
i

=
log


i / 2
2
1 e
k
k 1
 1
.320650948005153951322...   Li3    
 3

1 .320807282642230228386... 

2
coth 2 

.32093945142341793226... 

1

4

sin 2 x
dx
x
1
0
e
  (2k )   (2k  2)
k 1
2
log k
k
.320987654320987654
=
.321006354171935371018... 
.321027287319422150883... 
.32104630796716535150... 
.32111172497345558138... 
1 .32130639967764964207... 
26

81
2
dx
 (1  x )
4
0
e1/ 4

4

k
 k !4
k 1
200

623
k


Fk 4
k 1
8k

cot1

2
7 2 
3 log 2 2
  log 2 

48
4
4
4
5

log(1  x 2 )
1 x (1  x )2 dx

2
2
3


csc
5
5
5 5
dx
 1 x
5/ 2
0

4
2
 4

 2 log 3      ( k )  1 
 
3 3 3
k 2 3

1

( 1) k 1
erf  
2 k 1
2
3
(2 k  1)
k  0 k !3
k
2 .321358334513407482394... 
.32138852111168611157... 
.321492381818579142365... 
16

 37

tan

2
63
37



k
k 1
2

 9k
k 2
2
16
 12 k
1

 7k  3
 (k )
log ( k ) 
k
1
  (1) k 1
.3217505543966421934...  arctan  arctan 2    2 k 1

3
4 k 1 3 (2k  1)
.32166162685421011197... 
k 1
 2 

arctan
2 
k 1
 (k  1) 

1


arctan

2
 2 ( k  1) 
k 1


 
=
=
 (1)
2

 
=
x
1
1 .32177644108013950981... 
1 .321779532040728089443... 
1 .321790572608050379284... 
k 1
[Ramanujan] Berndt Ch. 2, Eq. 7.5
[Ramanujan] Berndt Ch. 2, Eq. 7.6
dx
1
2

1  1
HypPFQ{1,1,1},  ,2,  
2  4

 sin 1

  2k 
k 0
1  dx

   cos x  cos 
2
x  1  x2
0
2 ( 3)   ( 4 ) 
1
1
  ( k  1)
k


2 .32185317771930238873... 


1
 2
i 1 j 1 k 1
log5
.321887582486820074920... 

5
2 .321928094887362347870...  log 2 5
3 .321928094887362347870... 
.32195680437221399043... 
1 .321997842709168891477... 
ijk

Fk 2

k
k 1 4 k
log 2 10

tanh
 3

31

21

k
2
1

 5k  7
2
3
k 1
1
csch 2 sin   1  i sin   1  i
2


 
   kk

k 1
 2k 2  2
4
 2k 2  1
4
1 .32237166979136267848... 


2


5 5 5 5 2 5

5/ 2






 5 47  21 5  2 5  2 5 75  33 5  2 11  5 5 arc csc h2  




 
=
Fk 2

k 1  2 k 
 
k
2

1
1
.322467033424113218236... 
   2
12 2
k  2 2k
2
1

=
  Li2 ( e2i )  Li2 ( e 2 i )
24 2

Hk
= 
k  1 k ( k  1)( k  2 )

=
k
 (1)
k 1
k 2
=
.322634...

  ( k )   ( k  1)

 1



2
 (1)
cos2 k
k2

 0 (k )  (k )
k 1
k2



2 .32272613946042708519... 
1 .322875655532295295251... 
2
7
2
e
2
1
e
2
1

Borwein-Devlin p. 90
2
14 .32305687810051332422... 
2 I 0 ( 2 ) 
e
2 cos x
dx
0
.32314082274133397351... 
 1  5 (3)
2 Li3   

8
 2
.323143494240176057189... 
 ( 2 )  2 ( 3)   ( 4 ) 
.323283066580692942094... 
( k  2 ) ( k )
k!
k 2

H ( 2) k

k
k 1 2 k ( k  1)

k2

4
k  1 ( k  1)


15 
 7
   
 2
8

H log k
2 .323379732200195668706...   k
k 1 k ( k  1)
3 .323350970447842551184... 

1 .32338804773499197945... 
9 sinh 

25
.3234778813138516209...

.323712439072071081029... 
6 .323732825290032637102... 

 
=
.32378629924752936815... 
3

k 1
2



k

 
log 2 5

8

 1 
k 1

1

sin   1  i sin   1  i csc    1  3  csch   1  3  




4
2

k  2k  2

4
2
k 1 k  2k  2

3 .3239444327545729800...

(1)
k
1
k 0

1
k
3 log 2  log     2   ( 1) k
 ( k )  1

2
k 1
k 2

1
1
sinh  
AS 4.5.62
2 k 1 

k  0 ( 2 k  1)!

.323409591142274671172... 

 1  ( k  3)
log(1  4 x )
dx 
1  4x
0
1

Hk 

2k 
arccsc 
1

.324027136831942699788... 
2 cosh 1
.323946106931980719981... 

 (1) e
k
k 0
 ( 2 k 1)
J943

H H
11 (3) 17 4 49 2
4



  k 2 k 3
3
360
216
9
k
k 1
2

1
cos k

 1
i
i
.324137740053329817241... 
 
  Li2 ( e )  Li2 ( e )   2
6
2 4
2
k 1 k


 (2k )  1
1
1
.32418662354114975609...  
sinh
2 k 1  
2k
k  1 ( 2 k  1)! 2
k 2 k
6 .324033241647430377432... 
.32420742957670372545... 
 15
30
491

770
cot  15 

k
k 1
2
GR 1.443.4
1
 8k  1
log(1  x ) log x
dx 
x 3/ 2
0
1
4 .324426956609796143497... 
6 .324555320336758664...

1 .3246052049335865...


 2  8 log 2 
40  2 10 

(1) k

k  2 ( k  1)( 2k  3) log k

1 .324609089252005846663... 
3 .324659569140513181898... 
1 .324666791899989156496... 
.324686338305007385255... 

 
=
cosh

4



1
 1  4(2k  1)
k 0
2




 (k ) log k
k 1
k


1/ 3
1/ 3
1 9
69 

 2 / 3  
1 .324717957244746025961... 

3 2
2 
3
 
= the plastic constant, unique real root of x  x  1  0

2 .32478477284047906124... 

J1079

k2  3
3
sinh  3 csc h  2   2
2
k 0 k  2
2

 (  1)
k


3
k
(  1)
k 1 
6
2
12 2 (2)  144 (2)   12 2 (2)   4 StieltjesGamma(1) 
6
1  27 3 69 
 

3  2
2 




t3 ( k )

k
k 1 2


 0 (k )
2
k 1
k
1


  
1
1



jk
 1 i 1 j 1 k 1 2ijk
j 1 2
 
k 1
.32525134178898070857... 
( 1) k 1 ( k  12 )!

k ! 2k
k 1
3 6
 
3



.325322571142143252138... 
.325440084011503774969... 

4
2 1 
 2 (k )
2
k 1
k
1
1
e
2x
0
dx

 e 2 x

1
3
.325735007935279947724... 
.325921357147665220333... 
4
0
arctan e2 


2
8

8 .32552896478319506789... 
  x(1x x1) dx



 (2k )  1
k 1
( 2 k )!



1

  cosh k  1

k 2
 3i 7 
  3  i 7   
7
4  
  Li2 
 
2  2  log  arccsc 2 2  i Li2 




4
4
2
7 7  



 

Hk
= 
k 1  2k  k
 2
k 
.326149892512184321542... 

 
.326190726563202248839... 
1 .326324405266653433549... 

 (k )
k 1

k!

 (1)k 1 22 k 1
k 1
 (2k )
( 2 k )!


 1
 sin  k  
2
k 1


.3264209744985675...

1 .32646029847617125005... 
H ( 2 )1/ 6
224
 128 
3


1  2 k  2

k 
 k 
 16
k 0

cos 2 x
.32654323173422703585...  cos 1 
 si (2)   2 dx 
2
x
1

3
k
 (k  1)  1
.32655812671825666123...  4    log 2  6 (1)  
2
k 1 k  3


( 4 k  2)
k2
1 .32693107195602131840...  


4
2k
k 1
k 1 2 k  1
 ( 3)
3 .326953110002499790192...  e
2
43 .327073280914999519496... 

imaginary part of zero of  (z ) 
9 .3273790530888150456...

.32752671473888716055... 
87
17
 2 (2)  6 (3) 
4
1
(   log 2  log 3) 
3
4
3 .327629490322829874733... 
 ( 3)
.3275602576698990809...



1
log 2 x
dx 
x 3 ( x  1) 2

 (1)
k 1
k
 (k )
2k
 ( 1) k 1
( 1) k 1 
3 .32772475341775212043...  8G  4   

2 
( k  1 / 2) 2 
k 1  ( k  1 / 2)


3
4
k 1

1
3
(2)  1 
129 .32773993753692033334...  2  56 ( 3)       2 
1 3
 4
k  0 ( k  4)

1
k
.327982214284782231433...   2
log
k 1
k 2 k  1
1
.32813401447599016212...  1     (1  i )   (1  i )
2
7 .32772475341775212043... 
8G 
k
2
k


  k  1, 
Adamchik (28)


 
=
 ( 1)  ( k )   (2k  1) 
k
k 2

 1
 1 
e  I 0    I1     1 
 2 
  2

1
F  , 2,1  1 

2

( 2 k  1)!!
 
= 
k 1 ( 2 k )!!( k  1)!
 4 
256
2 .328209453230011768962... 
 

35
 1 / 2
.32819182748668491245... 
1 .32848684293666443478... 
1 1
3 log 2
3


2
16
 /2
=

0

.328621179406939023989... 
.328962105860050023611... 
 (k )
2
k
k
3 2
2
 2 log 2 2  2 
3
 e e
x
0

2
k 0

=
arcsin 3 x
0 x 3 dx 
x 3 cos x
dx
sin 3 x
k 1
13 .32864881447509874105... 
1
x dx
x
 1/ 2

k
1
(k  2)2

1
 (k  1)!4
k 1
k
 2k 
  
k 
.328986813369645287295... 
2
30
 /2
1
.32923616284981706824... 

2 2 log 2  7 (3)     x log sin x dx 
16
0
log(1  xy )
dx dy
2 2
0 0 1 x y
1 1

 
=
1 .329340388179137020474... 

3 
5
   
4
 2

e
 x2
(1  x 2 ) dx 
0


1 .329396468534378841726... 
log k
 (k  2)(2k  3)
k 1
|  ( k )|
4k
k 1
1

(1  2   2 log 2  2 log 3) 
9

.32940794999406386902... 
.3294846162243771650...
3 .329626035009534959936... 
3 .329736267392905745890... 

  3log 2

2
2 2 39


3
12
 (1)
k 1
k
 ( k )k
2k
x
1
log log dx
x
 log x
 
0
GR 4.325.11
(1  x ) 2 log x
0 x  1 dx
1

1

( 2 k  1)
k 0


k
k
.329809295380395661449...   k k
 2  k
k 1 2 ( 2  1)
k 1 2  1
.329765314956699107618... 
arctanh
1
1


2
k 1
2 k 1
k (k  3k  2)
2


p prime
1
p( p  1)



 
 3 csch 2
  2 2 coth
csch 2
 2 coth

16 
2
2
2
2


2 k 2 (2 k 2  1)
k 2  (2 k )
 
=
(

1
)
k



2
3
2k
k 1
k  1 ( 2 k  1)
 2 1036
 7
.330357756100234864973... 

  ( 1)  
 2
2
225
.33031019944919003837... 


 (k )

.3302299...

 

h( k )
where h ( k ) 
3 .330764430653872718842...  2e  Ei ( 1)   1  
k 1 k !


1

1 .33080124357305868479...    1 

( k  1)( k  6) 
k 1 
k
H
i 1
i

23 .330874490725823342456... 
.330893268204054533566... 
20 .33104603466475394519... 

 

 
=
1 .331265897019480833836... 
1 .331335363800389712798... 
31 .331335637532090171911... 
.331746833315620593286... 
.33194832233889384697... 

45 (5)
x 4 dx
log 4 x
  x
  2
dx
2
e 1
x x
0
1
 log( e  2 )
1
cosh(1  e)  sinh(1  e)  cosh(1  cosh1  sinh 1)  
2
1
 sinh(1  cosh 1  sinh 1) 
2

e 2  e  e1/ e
sinh k
 
2e
k 1 ( k  1)!

27 log 3 27
1

 
Prud. 5.1.25.20
2
2
2
k 1 k (k  1 / 9)
 1/ 4
 2
 3
  3,  
( k  1)( k  2)( k  3) ( k )
3k  4
k 4


1
cosh 1sin 1  cos1sinh 1 
2

4


4 3


 (x
2
0
(1) k 1 2 2 k 1 k


(4k )!
k 1


  2 log 2 

.33235580126764226213... 

0
Berndt 8.6.1
 ( 4 k  3)
k 1
1 .33271169523087469727... 
 sin x sinh x dx 
dx
 1)( x 2  3)
3 log 3  3
 1

   
 6
2
2


 ( 3k  1)
k
1 .33224156980746598006...  
  3 
k
2
k 1
k 1 2 k  1

3 
( 1) k 1 ( k  12 )! 3k
.33233509704478425512... 
 

16
( k  1)!
k 1
6 .33212750537491479243... 
1
4 4k 3  1
4 2 

.33302465198892947972... 
log 3 2
.333177923807718674318... 
 2

.333333333333333333333
=
1
3

=
1


k  1 k ( 3k  3)

1


k  0 ( 3k  1)( 3k  4)

1
 (2 k  1)(2k  5)
k 0

=
=

1
1
  2

2
k  1 k  5k  6
k 1 4 k  4 k  3
k 1
k


( 1)
k 1 2 k
  ( 1)

2k
 2k 
k 1
k 1
 
k
 (k )

=
3


1
k
k 1

 (2k )3k
9k  1
k 1

=
2k


k  0 k !( k  2 )( k  5)
k 1
k 1
 
k
  2k 
k ( k  3)

=

 ( k  1)( k  2)
J1061
k 1
2 

1 
 

k ( k  1) 
k 2 

=

=
=

k 1

x 5 dx

ex

 e
3


0
0



ex
3

1
x

2

x2
0 (1  x )4 dx
x 2 dx
4
  1  (2k  1)
dx
0 (1  x )4 
0
=

1

sinh x
dx
2x
0 e

 2k 
1
4
 1 
 
   2,    k
3
 2  k  0 4 (k  2)  k 

1 

=  1 2 k 
2 
k 1 

  3 log 2
 ( 2k )  1

.333550707609087859998...  log  csc

 

2
2k k
2

k 1
1
.33357849190965680633... 

k
 ( k ) 1 4  1
1 .333333333333333333333
=
.33373020883590495023... 
.333850657748648636047... 
.33391221206618026333... 
sinh 1  sin 1

6


 (1   )
2 

2

12 3

1 log 2


8
12
(1) k 1

k  0 12k  8

J1061, Marsden p. 358

2 .333938385220254754686... 

 ( k  1)
2k  1
k 1
1 .334074248618567496498... 


8
4  4
arctan 2 x
1 x 2 dx

1
.334159265358993593595... 
coth 5 

10
50
.33432423314852208404... 

3 log 2  2

G 
4
16
k
2
k 0
1
 25
2    4arcsinh1  6 log 2

1
=
 arctan x arcsin x dx
0

2k k 2
44 .33433659358390136338...  6e  
k 1 k !

log( k  2 )
2 .33443894740312597477...  
k2
k 1
 /2
1 .334568251529384309456...  e
2
3 .334577261949681056069... 
 2   2

.33491542633111789724... 
1 

  1  2k !
k 1

 41

1
2
 7k  2
41
k 1

e
k
.33516839399781105400... 
 e log( e  1)  e   k
e 1
k  0 e ( k  1)
.335158712773266366954... 
1 .335262768854589495875... 
tan
6
720

.335348484711510163377... 

1


40
k
2
 volume of unit sphere in R12
(2 k )
2k
 1 3
 1 1
1 .335382914379218353317...    ,     , 
 2 4
 2 4
k 1
10 .335425560099940058492... 
.335651189631042415916... 
3
3
log   2 log 2   

 (1)
k 2

=
k 1
1 .335705707054747522239... 
1

1 
  k  2 log1  2k  
2
e2
k 1
 (k )
2k 1 k


k2

k
k 1 6
.336000000000000000000 =
42

125
.336074461109355209566... 
46 2 log 2


75
5

1
 k (2k  5)
k 1
( 1) k 1

2
k  1 ( k  1 / 2)

(1) k 1 k
2 .33613762329112393978...  6  4G   2
3
k 1 ( k  1 / 4)
.33613762329112393978... 
4  4G 


log 2 k
2 .33631317605893329200...  

k  2 k ( k  1)


.336409322543576932...


 ( k  1)
k
2 k
 2
Li1  
 7
k 1
.336472236621212930505... 
3 .33656672793314242409... 
1 .3366190702415150752...

2
e
e 1
k 2

 1
1
 k Li  2k 

 2 Li3 (2) 
2

 (k ) 
2
k 1
 2 log 2
3

.336945609670253572675... 
.336971553884269995675... 
.3370475079987656871...


3

2 .337072437550439251351... 
 (2k )  1

log 2 x
0 ( x  12) dx
2 1/ k 2
e
0
(1  cos x ) 2
dx
2  sin x

 k2 1

 (2k )

k 1
( k  12 )!
1 )!( 3k  1)
2
 (k 
k 1
 (2k  2)!!  


 k

 2  4 log 2  3 log 3 
k 1
.3371172055926770144...

 /2

k  1 ( k  1)!
k 2

1
1
 
Li5     k 5
 3
k 1 3 k

1

4
.33689914015761215217...  1   
 2 log 2 
3 3

log 3 2
 1
 2 Li3    
3
 2
1/ 2 k 2
e
k2
 ( 2k )  1
 2k 
 
k 
4
.337162848489433839615... 
8 6  2

k 1
.337187715838920906649...   k
k 2 k
k 1
CFG D1

6
( 1) k 1 k ( k  12 )!( k  12 )!
 

( 2 k  1)!
25 5
k 1

( 1) k 1 ( k  12 )!( k  12 )!
= 
2 ( 2 k  1)!
k 1
.33719110708995486687... 

 

2
3
3 

sin 
  1  2 
2
2k 
 3
k 2 
k

(1)
.337403922900968134662...  ci (1)    
 ReEi (1)
k 1 ( 2k )!( 2k )
.337264799344406915731... 
1 .337588512033449268883... 
.337610257315336386152... 
.3376952469966115444...

2
2
1
log 2
 
   arccot x log x dx
48 4
2
0
4 

21

(sin x  x cos x) 2
dx
0
x9 / 2

1

 2 log 1 
 
 2e 
e
3
 log 2 
2

.33787706640934548356... 

 (2 k )
dx
1/ 2
x
1

 (2 k )  1
k 1
k ( k  1)

k 2  1
2
1 .33787706640934548356...  
 1    1  ( k  1) log 2  
k 
k  1 k ( k  1)
k 2 
1
 
=   log 2
2

2


1
dx
.33808124740817174589... 
 log 2  2   3
2   3
6
x  x2
k 1 k  k
1



AS 5.2.16

1 .338108062743588005178... 

k 2
ppf ( k )

2k
8
.33868264216941619188...  6   e 
e
.338696887338465894560... 


 1  dx
 cosh x  x
5
1

1
Bk k



2
( e  1)
k 0 k !
18   2
1

.33876648328794339088... 
   x log 1   log x dx 

24
x
0
1


 
=
log( x  1) log x
dx
x3
1

2
4 k 2  3k  1
8 .338916006863867880217...  1 
 2 ( 3)   k  ( k )  1  
3
2
k 2
k  2 k ( k  1)


2
.3390469475765505037...

 2  2   3 
  

e
 2
8
x 2 dx
1
x2
0

  (k ) (k  2)

 1 
2
 (k  1)
k 2 



 (2 k  1)
1 
1 
.339395839527289266398...  
   log 1  2 
k
4 k
4j 

k 1
j 1 j

.339369340124745379664... 
 
 2 
16
 

15
  1 / 2

761
1
.339732142857142857143... 
  2
2240
k  5 k  16

e
k 2  23k
.339785228557380654420... 
 
8
2 k!
k 1
.339530545262710049640... 
1 .339836732801696602532... 



22 / 3 
1  i 3
1  i 3
2/3
 (1)1/ 3  2 


 
1
2
(
)




3 
21/ 3 
21/ 3  


22 / 3

  2  22 / 3 
3
 

=
 4  (3k )  1)
k

k

k 1
.339836909454121937096... 
 (k )  1
.340211994871313585963... 
.340331186244460642029... 
.340430601039857489999... 


 2 (k )
k 1
k!
36
  (2)  12( (2))      (k )klog

2
6
.340505841530123945275... 


2
2
k 1
2
k
2
29
1 (1 )  2 
4 2
1
4 (2 ) g2
 4
   
 1   ( 1)   


 3
 3
9
27
9
9
2
(k  1) (k )


3k
k 2

=
 1  1  1
I1  2   
k
 k  k
k 2

 k !(k  1)!   
k 2
6 .340096668892171638830... 
4
4
arccsc3

.339889822998532907346... 
k 2
3

log x
dx
3
1
x
1
 
 2  2  2  
4 



0
x 4 (1  x) 2
dx
x8  1

1
 (3k  1)
k 1
2


.340530528544255268456... 
 (k )  1
 k !(k  1)
k 2
.340791130856250752478... 
.34084505690810466423... 
 1
Li4   
 3
 1

2

1 .340916071192457653485... 
1 .340999646796787694775... 
.34110890264882949616... 
.341173496088356300777... 
1
3 k
k
k 1

4
( 1) k 1 ( k  12 )! (   1) k


( k  1)!
k 1
1
Fk
k 1

2
k
1 5 3


3
27

1

2k 
k 1


 k  1


( 1) k 1 ( k  12 )! 2 k
 
( k  1)!
27
k 1
k 4

9
(1) k
 
k
16 e
k  0 k !2


.34121287131515428207... 
  (k  1)  1 log k
k 1
 5
1 .341487257250917179757...     
 2

k
k 1
1
5/ 2


1
( 1) k 1  ( 3k  3)



6
3
2k
k 1 2 k  k
k 1

.34153172745006491290... 
4 .341607527349605956178... 
6
1 .341640786499873817846... 
3

5
  1 1 / 3  
 2  (2) 2 / 3  4 / 3
1 
1/ 3 2 / 3
  2
.34176364520545358529... 
(1) 2 (3i  3 )  
3  1       
  2  
2
6 3 




3  3i 
 1 
 2  3 (3)  22 / 3  1 / 3   
 

 
6 3 
 2 


1
( 1) k 1  ( 4 k  2)
 
= 


6
2
2k
k 1 2 k  k
k 1


H
(1) k 1 H k
2

.34201401950591179357... 
 log 2 2   k k
 
12
k 1 2 k ( k  1)
k 1 k ( k  1)



80 .34214769275067096371... 
4e 3 
3k ( k  1)

k!
k 0


GR 1.212
log ( k )

k
k 2

.342192383062178516199... 


1 .342464048323863395382... 
J
k 0
2 .34253190155708315506... 
k
(1) 
e  e  

k ( 3k  1)
2k 4



3
2 
2k
k 1
k  1 ( 2 k  1)


 ( 4 k  1)
1



5
k
4
k 1
k 1 4 k  k
1 
1 
i 
1 
i 



     1 
   1 
   1 

   1 



4 
2
2
2
2 

2 .342593808906333507124... 
.3426927443728117484...

=
945 
 11
  
 2
32
52 .34277778455352018115... 
.34294744981683147519... 
3 G  3 3 2 log 2 105 (3)




4
64
32
64
1
arctan 3 x
0 x 2 dx 

 /4

 



0
.343120541319968189938... 
x3
dx
(sin x) 2
8 log 2   ( 3)  4 

 4k
k 1
2 .343145750507619804793... 
8 4 2
1 .34327803741968934237... 
3152
 96 (3) 
27

.34337796155642703283... 
sin x
 (x  1)
2
5
1
 k3
1
log 2 x
0 1  x1/ 4 dx
dx
0
(1) k 1
1 log 2


 

12
12 3 4
k  0 12k  4

.3433876819728560451...

(1) k
log3 x cos log x
 2
dx 
x
k  0 ( 2k )!( k  2)
1

.343476316712196629069... 


1  e e
( 1)k e k


e
k  0 ( k  1)!

1
  k 1
3
k 1 2
2

74
9 ( 3)

 12 log 2 


2
120
2
.343603799420667640923... 
1 .3436734331817690185...
24 .3440214084656728122...
12  6 cos1 10 sin 1  
e
1

 
=
1

  log 1   log 3 x dx 

x
0
k 3 ( k )

k!
k 2
2
11

.34460765191237177512... 


18 54

H ( 3) k
.34461519439543107372...  

k
k 1 4

16 .344223744208892043616... 
2 .34468400875238710966... 
.344684541646987370476... 

k
k 1
3
1
 3k 2
1
1
 e erfi

4
e
265e  720 

erfi e  
e




cosh k
 k !(2k  1) 
k 0
k
 k !(k  6)
k 1
2
.344740160430584717432... 
5 .3447966605779755671...

2
2
2 1

 2  
cos x
   cos x dx
0
 1  4 
 csc        
 5  5 
5


 
=
 log1  x  dx
5
0
.34509711176078574369... 

1/ 4
4e


 xe
x2
sin x dx 
0

9 sinh 4
 256 
.3451605044614433513... 
  1  4  
1190000 k  5 
k 

sin 1  cos1 
k2
  ( 1) k
4
( 2 k )!
k 1

1 e / 2 1/ 2 e
sinh k
1 .345452103219624168597... 
e e   

k
2
k  0 k !2
.34544332266900905601... 

.345471517734513754249... 
 ( (k )  1)
2
log k
k 2


1


 (2 k )  1
1
.345506031655163187271...   
cot
 
  2

k
2 2 2
2
2
k 1
k 2 2k  1


1


(2 k )
1
1 .345506031655163187271... 

cot
  k   2
2 2 2
2
2
k 1
k 1 2 k  1

1 .3455732635381379259...

 x( x  1) (2 x)  1 dx
1
 /2
.345654901949164100392... 

 log 2  2 G 
0
1

 
=
1
3
k2
k 1
.3457458387231644802...

GR 6.142

( 1) k  2 k  1



k 
k  0 ( k  1)!

1 

 1
 log 2  Ei     
 2

e

.346101661375621190852... 
2
arcsin x
dx 
1 x
0

x
1 I 0 ( 2 )


2
2e 2
.345814317225052656510... 
1
  dx

  K (x )  2 
0

.345729840840857670674... 
x cos x
dx 
1  sin x
erfi
1

3


 (1)
k 1
k 1
Hk
k !2k
1
(2 k  1)
 k !3
2 k 1
k 0
 2 x
0  x  dx
1
1 .34610229327379490401... 
1 .346217984368230168984... 
 ( 1) 3/ 4  sin ( 1)1/ 4    ( 1)1/ 4  sin ( 1) 3/ 4  

 5

1
1
tan
   2

2
5
5
k  1 k  5k  5

 (2)
 (k ) log k
.346494734701802213346... 




2
 ( 2)
k2
k 2
.346250624110635744578... 

3
4
(1) k k
 5

 erfi1 
.34654697521433430672... 
1F1  2, ,  1  
3 
 2e
 2
 k  0 (k  12 )!
log 2
.34657359027997265471... 
 Relog(1  i)   Re Li1 (i) 
2

1
1
1
  2 k 1
= arctanh  arcsinh
AS 4.5.64, J941
3
(2k  1)
2 2
k 0 3


 1 1
H Ek
=  k 1    Lik    
 2 2
k 1 2
k 1 
1
=

( 1) k
cos k / 2





k
k 0 2k  2
k 1
=




k 1

=
 (2 k )  1
2k
dx
 (x  1)(x  3)
1
1
=


k 1


x
1

1 
1

log 1  2 

2 k 2 
k 
dx
x
3
x 1
dx
2
 1) log x
 ( x  1)( x
0
 1   ( k ) 
GR 4.267.4

=

dx
0 e2 x  1 
e
0
 /4
=
 tan x dx
 /4

0

=
 (1  e
x
)
0
x dx
x2

0
1
cos x  sin x
dx
cos x  sin x
cos x
dx
x
1
=
   ( x ) sin  x sin 3 x dx
GR 6.496.2
0

=
x 2 sinh x
0 cosh2 x dx
GR 3.527.14

=
3  4 sin 2 x 2
0 x sin x dx
Prud. 2.5.29.25
4 

3 
.346645900013298547510... 
 cos  cos  
7 
14
14 

k
k 1
.34700906595089537284...   ( 1)
3
k 1
k 1

x 3 (1  x) 2
0 1  x 7 dx

.34720038895629346817... 
2
2 log 2  2 log 2  2 
2
k 1
1 .3472537527357506922...

.347296355333860697703... 
240 .34729839382610925755... 
 1
Li1/ 2   
 2
2 sin

18


2
k 1
k
k
2  2  2  2... 
log5 x
 
dx
4
( x  1)( x  1)
0
.347376444584916568018... 
32 log 2 
erf


k 2

2 .347560059129698730937... 
[Ramanujan] Berndt Ch. 22

6
.347312547755034762174... 
.347413148261512801365... 
Hk
(k  2)

2 4
24 (5)  12 ( 3) 

5
.347403595868883758064... 
k
131

6

2
k 0
1

 (k )  1
k
log k 
2k
 (k  1)  1 

k 1 k
k
GR 4.264.3

x4
0 (ex  1)3
1

( k  5)
.34760282551739384823... 
.347654672476249243683... 

 

 
=

0

i 3
21/ 3
1
( 1) 2 / /3  2  2 / 3  2 / 3  
3
2
2 

 
 ( 1)
k 1
k 1

.3477110463168825593...

cos2 x
dx
(2  sin x ) 2

i 3
21/ 3 
1
  2  21/ 3   ( 1)1/ /3  2  2 / 3  2 / 3   
3 
2
2 




 /2
4
 1
  
3 3 2 2
2  (3k )  1 
k

k
k 2
3
2
2
  (k ) (k  1)  1 
k 1
3
1 .348098986099992181542... 

23
2 2  4
.348300583743980317282... 

 2 ( 3)  16 log 2  16 
3
90

1
= 
5
4
k 1 2 k  k
1 .348383106634907167508... 
7 .3484692283495342946...

7 .348585884767866802762... 
.348827861154840084214... 
.34906585039886591538... 
.3497621315252674525...

 

 
2 
2
(1) k 1 ( k  4 )


2k
k 1

 i logi
2
54  3 6 

1
 cot 3 
3
 1
Li3   
 3


1
2
k 1

k
3

2k
k
3
7/2
x dx
 

9
1  x9
0


1
( 1) k 1  ( k  1)




2
2
4k
k 1 4 k  k
k 1
 1
 5
= hg         
 4
 4

4
 3 log 2 
=
  log(1  x 4 ) dx

1
0
.349785835986326773311... 

p prime
1 .3498073850089500695...
Berndt ch. 31
1
3 k
k 1
tan

1
2 1
p
 coth 2   
1
1   1  2i    1  2i   
2
4( k  1)
=  3

k 1 k  4 k


 
.349896163880747238828... 
k
k 2


 ( 1)
k 1
k 1
4 k  (2 k )   (2 k  1) 
1  5 1  2 
1 2

      ,1    , 0,1 

2  3 3  3 
3 3

.34994687584786875475... 

2
k
 ( k )  1 
1
1
e
0
x3
dx
.350000000000000000000 =
7
20

.350183865439569608867... 

1 

2k 
  1  2
k 0
 /4

cos2 x
1
1
dx 

arctan
 
2
cos
x
4
1

2
2
0

|  ( 2 k )|
.35024515950787342130...   k

k 1 4  1
2

log 2 log 3
x dx
.35024690611433312967... 


 3
x 1
3
6
6 3
1
.35018828771389671088... 

.350402387287602913765... 
2 .350402387287602913765... 
1
k  1 ( 2 k  1)!
2 sinh 1  2  2 
e

1
1
 2 sinh 1  2

e
k  1 ( 2 k  1)!


 
=
e
cos x
sin x dx
GR 3.915.1
0
1 .350643881047675502520... 
 1
E 
 2
141 .350655079870352238735... 
52e 

k 5
(2 k  1)2

4


k!
k 1 k !
k 1

Berndt 2.9.7

.350836042055995861835... 

 


 
=
 4  22 / 3  i 22 / 3 3  
21/ 3 
2/3
 1/ 3

  

 1  2   (1)  
6
6 
4


2
  ( 1)
1/ 3

 (1)k 1
k 1
1 .3510284515797971427...

.351176889972281431035... 
15 .351214888072621297967... 
21/ 3  4  2 2 / 3  i 2 2 / 3 3 


6 
4

 ( 3k  2 )
2
k


 2k
k 1

3
   k 2  (2 k  1)  1 
2
4
k 2
2
log 2 2



24
8
 ( 2 )  6 ( 4 )  6 ( 3) 

=
k  4k  1
( k  1) 4
k 2

2
4 k 4  3k 2  1

2
3
k  2 k ( k  1)

 (k  1)  (k )  1 
3
k 2
 
1
 k2
 (3)


5
Berndt 9.8
.351240736552036319658... 
.351278729299871853151... 



4 5
dx
2
5
 4x
0

k
 (k  1)!2
2 e 
k 0
 2
 
4 .351286269561060410962...  K 
 2 
2

.35130998899467984127... 
 2k
k 1
.35144720166935875333... 
16 
1
 K (k )
0
k
dk

k
GR 6.143

1
( 1) k 1 ( 4 k  1)


5
2k
 k k 1
 2  32
2
1
arcsin 2 x
dx
 
1 x
0

.351760582118184854439... 
kHk
k  1 ( k  2 )!
4  2 e  3  e  e Ei (1)  3Ei (1)  
 (k  1)  1

.351867223826221510414... 

k 1

.351945726336114600425... 
2k k

1
1

   log 1   

2k 
k 2 k
Ek
k 1 k!

9 3 81
1 .351958911849046348014...  3 


4
2
e2

 (k
2
2 .352009605856259578188... 
k 1
2
1
 1 / 9)2



1
 2 ( k )



k
2
4k
k 1 ( 4  1)k
k 1

3log3
1
1 .352081566997835463...
 2 
 k
2
k  0 4 ( 2k  1)( 2k  3)

1
1
1
.352162954741546234686... 
sinh
  k
3
3
k  1 3 ( 2 k  1)!
i
i


  ( siin1) / 2
cos cos1  i ei  ( i ) e
.35225012976588843278...  2 e
2


.3520561937363814016...

24 .352272758500609309110... 
4
4


1 .352314016054543571075... 
.352416958458576171524... 
.35248587146747709353... 
9 (3)

8

x3
0 cosh 2 x dx
1
SinhIntegral (3) 
3

2
28

 1  dx
2 3
x
 sinh x
1

( 1) k
.352513421777618997471...  arccot e 
 arctan e   2 k  1

(2 k  1)
2
k0 e

=
e
1
1 .352592294195119116937... 
.35263828675117426574... 
x
dx
 e x
1 2
 e    Ei (2)  log 2  1 
2
1
1
 e Ei (1)  Ei (2)  
2e

2k k

2
k  0 k !( k  1)
1
dx
 e (1  x)
x
2
0

2 .352645259278140528155... 
2 .352694261688669518514... 
.352809282429438776716... 
.35283402861563771915... 
2 .35289835619154339918... 
1 .352904042138922739395... 
.3529411764705882
=
.35323671854995984544... 

 

k3
54  19 e  
k  1 k !( k  3)
2


(1) k 1
 6  32   2
3
2
k 1 (k  1 / 4 )

(1) k  ( k )

k
k  2 2 ( k  1)


( 1) k
k3
k
J 2 ( 2)  
  ( 1)
( k !) 2
k  0 k !( k  2)!
k 0


 ( k  1)
1
1

Li2


2
k
k
k 1
k 1 k
2


H
5 ( 4 )
1

 5 ( 4 )   2 ( 2 )   3k
72
4
2
k 1 k
6
17
2

 
 

k
1  p   
  1   1  
p prime 
k 1

LY 6.117
Berndt 9.9.5
probability that two large integer matrices have relatively prime
determinants
Vardi 174
.35326483790071991429... 

2
1
( J 0 (1)  cos1)   sin x arcsin x dx 
0
( 1)k 1 k 2
 
2k  1
k 1

.3533316443238137931...
1
.35349680070142205547... 
x
1/ x
dx 
0
1 .353540950076501665721... 
.353546048172969039851... 
 ( 3) ( 2 )
cos1 sin 1


2e
e


1
x sin x
dx
ex
.35355339059327376220... 
.353658182777093571954... 
.35382477486069457290... 

3
2
8

 sin sin
8
8
4
8
2
 1 

 
 coth  
4 2
6
2

log
csc





1

4
2 
k 1 k  k

x
1
4
dx
 x2
 ( 2k )
5k k
 1
 1
.353939237813914641441...  log( e  1)  1  2 Li2    2 Li3    2 ( 3) 
 e
 e
1

 
=
5
k 1

Bk
 k !(k  2) 
k 0
2
x dx
x
1
e
0
5
 2
  
 3


(4 k )
1
.35417288226859797042...  


4
k
4
k 1
k 1 4 k  1
1
 

 

 coth
=
 cot

2 4 2
2
2

k  (k )

.354208311324197481258...   ( 1)
k !k
k 2
1 .354117939426400416945... 
3 7
1
.35429350648514965503... 
1  i 3  1  (1)1 / 3 22 / 3  1  i 3  1  (2) 2 / 3  2 1  22 / 3 
1/ 3
12  2

1
 
=  3
k 1 k  4
.354248688935409409498... 



 


2
1
1
2 3
 log   k

arctanh
3
3
3
k 1 3 k ( 2k  1)

 2 k  ( 1) k 1
1  1
.35496472955084993189...  1 
I0       
e  2  k 1  k  k !4 k
.354880888192781965553... 
.35497979441174721069... 
 

 

 
2 
 2  1  log
 
2
1 2 


1
 arctan x arccos x dx
0


2   (2)  Li2  e  Li2  e 2i 


1
4k  1
= 



2
2
2
k 1 k ( k  1)
k 1 2k ( 2k  1)



1




(
k

2
)

1

(1) k  (k )   (k  1) 
=  3


2
k 2 k  k
k 1
k 2


( 1) k k ( k  1)
k 1
=  ( 1) k  ( k  2)   

2k
k 1
k 2
.355065933151773563528... 


2i



1
=
 log x log(1  x ) dx
GR 4.221.1
0
.355137311061467219091... 

4 .355172180607204261001... 
7 4

1920
log 3 x
1 x 3  x dx 
2 log 2 

=

x dx

GR 3.452.1
ex  1
0

=
log( x 2  9)
0 x 2  1 dx

=
x sin x dx
 1  cos x
GR 3.791.10
0
2
=
 x
  log sin   dx
 2
0

=
  log 2 (1  cos x ) dx
GR 4.224.12
0
 /2
=

 log (cos
2
0
1 .3551733511720076359...

.355270114150599825857... 
.35527011687405802983... 
5

.355379722375070811280... 
.3557217094508993824...

.35577115871451113665... 
1 .35590967386347938035... 

3 1
6 2
3
 log  
2
x ) dx

GR 4.226.1
dx
 1 x

12 / 5

0


 (2 k )  1
k 1
k 1
( 3)   ( 2) 

3
tanh
1 


    k 2 log 1  2   1

k  
k 2 
 3
2



k
k 1
5
1
 k4  k3
(k!)  (2k )

(2k )!
k 1

.3553358665526914273...

2

2
1  2 2 / 3 2 2 1/ 3
3
e
 e
cos 1/ 3 
3
3
2
 (6)
 (3) 

 (3)

1

4
k  2 k  13

1

1  k 

4 
k 1 
(1) k 4 k

k  0 ( 3k )!


.3560959957811571219...

  coth k  1
k 1
2 .356194490192344928847... 
=
1
3
 arccot(1)  arctan  2 arctan 2 
4
7

2
arctan 2 

k
k 1
K Ex. 102
[Ramanujan] Berndt Ch. 2


 
=
.356207187108022176514... 
.356344287479823877805... 
.356395048612456272216... 
.356413814402934681864... 
dx
 2x  2
0
log 7 2


 1
 (2k )
coth    (1) k 1

4
2 2
4k
k 1
k
1
3 1 

F
2
,
2
,
, 

2 1
4
2 8  k 1  2k  k

 2
k 

 (k )

k 
k 1 4
x
2

 4k
k 1
1
2
1


.3565870639063299275...

.356870185443643475892... 

 (1) k 2 
( 12 )

  1 
3 
( 2 3 )( 16 ) k 1 
4  1
Li   
3 2  4
2
.356892371084866755384... 

1 
1
H (2)k



k
k
3 k  0 4 ( k  1) 2
k 1 4
dx
  ( x)
1
.3571428571428571428
=
5
14

.35738497664673044651... 

1 .3574072890240502108...

3
k 1
. 35776388450028050232... 
6 .3578307026347737355...

 


2 1  1 
  
3
 2

e
0
x dx
x3
1
2
k

1
k
6G  1

4
2
 2k 
1
 


k
k  0  k  16 ( 2k  1)( 2k  3)

3 3
27 log 3
 24 log 2 

2
2

1
= 

1
1
k  0 ( k  1)( k  2 )( k  3 )

J1028
.357907384065669296994... 

1
2
1  cot 1 
k 1

=

k 0
.358145824525628472970... 
.358187786013244017743... 
1 .358212161001078455012... 
k

sin


k4

k  0 ( k  3)!

2

1  e  
2
2
 sinh 1
e
 sin(2 tan x) cos x
61

128
7

4 .358830714150528961342... 
  3
(k )
k 2
.358832038292189299797... 
 
=
  sin( 2 tan x)
0
dx

x
 /2
 sin(2 tan x) cot x dx 
0
6
GR 3.523.9
 3
8k
19
  1 2 / 3 
1 
1 / 3 2 / 3
 1  i 3  1  (1) 2
  1  i 3  1       2 1  2 
1/ 3
  2 

12  2 



 4k
k 1
k
3
1

 


5 .35905348274329944827... 
17 4  2

 2 (3) 
360
6

1 .3590982771354826464...

e
dx

x
x
0

e
0
1 .359140914229522617680... 
 
1 x
 xe dx 
2
=
0


 
=

k 1
H k H k 1
k2
e
k

1  
2
k  1 k !( 2 k  4)


e
k
k ( k  1)
 
 

2
2k !
k  1 ( 2 k  1)!
k 1



x dx

x
x

.359140914229522617680... 
dx

x
0
k 1


x

 cosh x dx
 2 (2k )
4 .358898943540673552237... 
.358987373392412508032... 



1 
2

k 1
0
1439 .358491362377469674739... 

 1  2k

=
Berndt ch. 31

195
36e 

2
2
1

2k

 2k 

1
  csc 2
tan
k
1
dx
e (1  x ) 3
0

x


e

 
=
log x
 (1  log x)
2
dx
GR 4.212.7
1
.359201008345368281651... 
2 .359730492414696887578... 
3 ( 3)  3 ( 4 )
 3/ 4 

3 .35988566624317755317... 

1


k 1 Fk


k 1
 (k )  1
k

1
 ( ) k
1
1 
k!
k
k 2
k 2

sin 2 k
.359946660398295076602...  arccot (2  cos 2) csc 2   k
k 1 2 k
3

34
Fk
1 .360000000000000000000 =
  (1) k 1 k k 
25
2
k 1



 ( 4 k  2)  k 2
tanh   (1) k
.360164879994378648165... 
 4
8
2
4k
k 1
k 1 4k  1
.359906901116773247398... 
1 .360349523175663387946... 
1 .36037328719804788553... 

 3
4




  e
1/ k
1
 (2k  1)
k 1
2
51840
 77
cos

4199
2


k 1

1   ( 2 )   ( 3)   ( 4 )
.360816041724199458377... 
6
4

e
1 .361028100573727922821... 
1

 1 k (k  9) 
.360553929977493959557... 
.361028100573727922821... 
GR 1.421
 1/ 9

(1) k

k
k  0 k!2 ( k  2)

3
1

coth  2    2
4
2 2
k 1 k  4k  6

1

1

coth  2   2
4 2 2
k 1 k  2k  3


13
1
  2
36
k 1 k  5k  4
49
 H ( 2)3
1 .361111111111111111111 =
36

dx
2
.361328616888222584697...   e Ei ( 2 )   x
e (x  2)
0
1
.361367123906707805589...  arcsin
2 2

1

1
.361449081708275819112...  
 coth 4   2
32 8
k  1 k  16
.361111111111111111111 =
J124

3
3
kH
2 3
log
log



  k k
2
2
2
k 1 3 ( k  1)
3
2
2
2
  (6    ) log 2 3 log 2 2 log 3 2
5 .3616211867937175442...  




 (3) 
2
4
4
4
2

log 3 xdx
 
=  
e2 x
0
.361594731322498428488... 


k7
441 .361656210365328128367...  162 e  1  

k  1 ( k  1)!
 
log k
1 .3618358603536914...
  n
k 2 n2 k  1
3

e 1
3k
6 .3618456410625559136...  
 
3
k  0 ( k  1)!
1
.3618811716413163894... 

2/3
12   ( 2)    2 2 / 3  ( 1)1/ 3 2 2 / 3 
k3  4

k3
k 2


 2k 
38
1
    k
105
k  0  k  4 ( k  1)(k  4)

sin k
.3620074223642897908...   
k 1 ( 2 k  1)
.3619047619047619047
=

5 3 39 (3)
(1) k 1
.36204337091135813481...  27 

 
3
4
6 3
k 1 ( k  1 / 3)

1 
 1
k 1  (2 k )
.362131615407560310248...   ( 1)
Li2   2 

2  
(2k )
4 k 1  k 
k 1

1
 2 
.362161639609789904943...  I 0 
 1  
2 k
 3
k  1 ( k !) 3

.36219564772174798450... 

 1  2(2
1
k


 1) 


 ( p(k ))
(1) k 1
.3623062223664980488...   
 
, p(k) = product of the first k primes
p(k )
k 1 p ( k )
k 1

1
.362360655593222140672...   k

k 2 2 (k )
k 1



5 (3)  2 log 2
(1) k 1 H ( 2) k

 
4
6
k (k  1)
k 1

1
.362532425370295364014...  
k 
k  1 ( 3k )


29 
( k )
 ( 2 k  1)  1
.3625370065384367141...  

  ( 1) k k 1  
15 2
4
16 k
k 2
k 1
.362389718306640099275... 
1
2
.362648111766062933408...  erf

3

.3627598728468435701...  
.3632479734471902766...

2 .363271801207354703064... 
.363297732806006305...


( 1) k

2 k 1
(2 k  1)
k  0 k !3


5 3
log( x  2)
dx 
x2
0
1
1
(log 2 3  log 2 2) 
2

GR 4.791.6
4 
 3
    
 2
3

sin ( 3 ) sinh ( 3 )
9


1  4 

2
24
k 
2 

log(k  1)
.363340818117227199895...  

ek k
k 1

1
 e  1  dx
 log e  x  log x
GR 4.221.3
0
2
1
 (2 k  1)!!
.363380227632418656924...  1 
 


k  1  ( 2 k )!!  2 k  1

2

=
 (1)
=
4
11

.363715310613311323433... 
1
 k !k ! 1
k 2

1 .363771665261829837317... 
1
F
k 1
k2
  7 ( 3)
2
.36380151974304965526... 
.363843197059608668147... 
 ( 2 k  1)!!

 ( 4 k  1)
 ( 2 k )!! 
3
k 1
k 1
.36363636363636363636
4
3


 ( 1) k k ( k  1)
k 2
17e1 / 3

9

4k
 (2 k  1)
k 1
1
dx
 4


4
3
x  x3
k 2 k  k
2
(1) k 1


2
k 1 k  1

sin x
dx 
x
1
e
0
2
sin x
dx
2
x
 cosh
0




1


cot  7 
.364021410879050636089... 
14 2 7

2
k

k3

k
k 1 ( k  1!3

1


2 2 sinh 

( k )

3
.363975655751422539510...   (3)   ( 2)  log 2 

2
.363985472508933418525... 
J385

k
k 1
2
1

 6k  2
3
1 .3641737363436514341...

35 (3)  2 log 2  2


G 
16
8
4

.364216944689173976226... 

  2k 
k 0
2k
  (k  1)
k 

2
Hk
  2k 
k 0
  (k  1)
k 


1 

7 .364308272367257256373...   1 
(k  1)! 
k 1 
2 .364453892805209284597... 
1 
erfi 2 
2 2
2 .364608154370872703765... 
2 CoshIntegral (2)  log 2    
.36481857726926091499...  1 
.36491612259500035018... 
2 .36506210827596578895... 

 

2k

k  0 k!( 2 k  1)

 ( 1)
k 1

4k


k 1 ( 2k )!k
 ( k  2)

3 67
3

e 
2
2
2

log


e1 x log x dx




4
0
1 .365272118625441551877... 
2   csch


1 
1

2 .365272118625441551877... 
1 
3 (3)
.365381484700719249363... 
.365540903744050319216... 
26 .36557382045216025321... 
.365673715554748436672... 





k
k 1
2
 1  i 3 
1   1 i 3 
  Li2 

   Li2 



27
3  
2
2




3
64 8
64


log 2   x 1/ 2 Li2 (  x ) 2 dx
9
27
27
0
2

k 1
(1) k


2
2
k 0 k  1/ 4

1
1 (1)  1
 3
    3
2 
 3
3
k  1 ( 3k  1)
k 1

( 1) k ( k  1)
= 
3k
k 1
3 .36513890066171554308... 
.3651990418090313606...

log 2 


2
2

CFG A22

1  4
4
 7 
  Root[3534848  152832#1 #13 &,2]   ( 3)     ( 3)    
6561 
9
 9 



 
=
.365746230813041821667... 
1 .365746230813041821667... 
log 3 x
1 x 3  1  x 3 dx

 (k )  1

 (k )

 1  1   1 
  1   log1   



 k   k 
k  2 k ( k  1)
k 2  k
 k (k  1)

k 2

 
=
.36586384076252029575... 

k 2 j 2
2 .366025403784438646764... 
=
 (k )
k
j

log k

k  2 k ( k  1)
  

 1  1   1 
1     1   log1   
 k   k 
k 2  k

1
k2
log

2
k
k 2 k  1
(1) k 1 (2k )



2
k 1 ( k  1)!( 2k  1)

1 .365958772478076070095... 


3

3 1
cos

 sin

1
1
 k erf k
k 1

6
6
2 

k
=   1  ( 1)

6k  3
k 1 

 (3k  1)
.36602661434399298363...    3k 1
1
k 1 2
1
1  ( 1) 3/ 4
3G i
.366059513169529355217... 
log

 2iLi2  ( 1) 3/ 4  
3/ 4 
4
1  ( 1)
2
96


.366204096222703230465... 
log 3

3

21 / 3

313 / 6

.366210241779537772398... 
.3662132299770634876...

=
.36633734902506315584... 
O
H k
k
k 1 4
x
0
dx

6
3
1
 2
Li2     (2)  log 2 log 3  log 2 3  Li2   
 3
 3
k 1


(1) H k
1



k 2
2k k
k 1 3 k
k 1

8k  1
 2 7
3 log 2  G  

 
2
2
4
9
k  2 k ( 4 k  1)
J1029
 /4
 x sec x dx
0
( k  1) ( k  1)  1
4k
k 1

=



.36651292058166432701... 
 log log 2
.366519142918809211154... 
7

60


sin x  cos x)  dx 
4
x6
0
371 4
 .366563474449934445032...  2( 1)
Li4   ( 1)   Li4  ( 1)   2 Li4   ( 1)  

9720

1  ( 3)  1 
log 3 x dx
( 3)  5  
 
=

          3
 3
 6  1 x  1
1296 
2/3
8 .3666002653407554798...

138 .36669480800628662109... 
3 .36670337058187066843... 
.36685027506808491368... 
2 .366904589024876561879... 
4 .366976254815624405817... 
.367011324983578937117... 

2/3
1/ 3

2/3
70
1160703963

8388608

k9

k
k 1 9
( 3)  2 ( 4)
1
2 1

16

4
 x arcsin
2
x dx
0
 1 3 
 1 1
(1) k

    ,     ,    2  

2k  1
 2 4 
0
  2 4
8
2
 4 log 4   (3) 
k 1
.367525688683979145707...   3

k  2 k log k

.367879441171442321596... 

( k )
k 1
2k


 (1)k 1
 (k  3)
k
(1) k k 3


k!
k 0

3
  ( s)  1 ds
2
1
 i 2i /   (1,1) 
e

=
dx
x cosh x

0
4
G
3
2
k 1
2

cos k
 ( 3) 1
.367042715537311626967...  

  Li3 ( e2i )  Li3 ( e 2i ) 
3
k
2
4
k 1

 ( 2k )
 (k )
.367156981956213712584...   2  k
  (1) k 1 k
2 1
2 1
k 1
1 .367630801985022350791... 

(1) k


k!
k 0

(1) k k 4


k!
k 0

(1) k


k  0 ( k  2)!


 2k
k 1
4
1

 k3


=
1
 (2k  1)!(2k  3)
k 0

=
 (k )
e
k 1

=
k
1
1
 StirlingS1(k ,1)
k 1
k ( k  e)

=
 (k  e  1)(k  1)
J1061
k 1


 
=
=
1 .368056078023647174276... 
dx
1 e x 

2
dx 
0
1

0
1
 1  dx
 x sinh x dx   sinh x  x
2
3
3
log(1  x 2 )
0 x 2 dx
1
 4 log 2 2 

.368120414067783285973... 
log x
 ( x  e)
 (1)
k
log ( k ) 
k 1

sinh 2 e 2  e  2
2k

 

2
2 2
k  0 ( 2 k  1)!

2 

=  1  2 2 
 k 
k 1 

log 2 
 3
(1) k
.368325534380705489...

 sech
 
2
3
3
2
k 2 k  k
7
.368421052631578947
=
19

 (2)
 (k )
1 .368432777620205875737... 
  3
 ( 3)
k 1 k
1 .3682988720085906790...

2k k


k 1 ( 2 k )!

Titchmarsh 1.2.12


1
1 

p ( p  1) 
p prime 
1 ( 2)
( 2)
.36855293158793517174...  Re  (i ) 
 (1  i)   ( 2 ) (1  i) 
2

 1
1 

=  

3
(k  i )3 
k 1  ( k  i )
=


.368608550929364974778... 
2
16



log 2  2 log 2 7 ( 3)



2
16
16


Hk
 ( 4k  2 )
k 1
GR1.411.2
2

.36873782029464990409... 
.3688266114776901201...

1

 3k 
k 1
k! 
k


1
(1  J0 (1))   sin x arccos x dx
2
0


3  3
1

coth

  2
2
2
3
k 0 k  1/ 3


log 2
 ( k  1)
.369669299246093688523...  1 


  ( 1) k 1
k 2
2
2
k 1

1 
 1

= 
 k log 1    1

k 
k 1  2 k
4 .369280326208524790686... 
.369878743412848974927... 
12 .369885416851899866561... 

3 3

2
6



4
k
 2
   ( k )  1 

 
k 2 3
4
 log 3 
3

 4 (3)   
15
k!
2 .36998127831969604622...  
k  0 S 2 ( 2k , k )


 9k
k 2
2
4

 6k
4
 k  1  (k  1)  1
k
k 1

.37024024484653052058... 
.370408163265306122449... 
.3704211756339267985...
.3705093754425278059...


2 .370579559337007635161... 



x 8 dx
0 1  x12
x 2 dx
 

1  x12
6 2
0
363
H
 7 
980
7

 (3k )
 k

k 1 3  1

1

 7k
k 1

1


3k
3
k 1 ( 3 )
k
3 log 3



4
4 3

1
3  4  
4
e 
2

1
3
k 0
3k
1
log(1  x 6 )
dx 
 

2
x3
k 1 6k  2 k
0

1
 /2
 sin
2
(2 tan x) cot 2 x dx
0

2k
3 .370580154832343626096...   k
k 1 k


1

1 .37066764204583085723...    1 

( k  1)( k  5) 
k 1 
3 .370690303601868129363... 
.3708147119305699581...

  sec
 5
2
 5 
 1 
 2





 1  k
k 2


2
1


 k  1
 log1  4( x
0
1
2

 dx
 1) 
GR 3.716.11
3
sinh 3 
2
2 .370879845350002036348... 
.37122687271077216295... 
.3713340155290132347...

=
2 .3713340155290132347...

4G
2

3k k

k 1 ( 2k )!
1

    x   sec x dx 
2
0
1

 sin  2   2

k  (2k )  1
1 
 csc 2


2



2k
2  8
2
k 1


2k 2


2
2
k  2 ( 2k  1)
 sin  2   2

1 


csc2
2  8
2

k ( 2 k )
2k
k 1



=
4 .37151490659152822551... 
.37156907160131848243... 
=
2k 2

2
2
k 1 ( 2 k  1)
2
4
 2 (3) 
1

2
 log 2

H k H k 2
 k (k  1)
k 1
( 1) k 1 Hk 1/ 2
G
 
4
2k  1
k 0
k

(4  1) (2k )
  2k
k 1 4 ( 2k  1)

Adamchik (25)
1
=
log(2 x)
dx
2
x
1

0
GR 4.295.13
log(1  x 2 )
dx
1  x2
0
GR 4.295.13
log cosh( x / 2)
dx
x
cosh
0
GR 4.375.1

1
=


.3716423345722667425...
=


1  ( 2 )  log 2 
log 2 (1  x )
=  2
x ( x  1) 2
0
1
4  (cos 2  1) log(4 sin2 1)  (  2) sin 2 
.371905215523735972523... 

4

cos2 k
 
= 
k  1 k ( k  1)
1

15 log 2 2 2 log 3 2  ( 3)


6
3
2
.37216357638560161556... 

1 4 log 
1
4


arccsch 2  
5
5 5 5
5 5
e

erfi (1) 
.3721849587240681867...  1  
2
4

k 1  ( k )
.372348858075476199725...   ( 1)
2k  1
k 1
.37236279102931656488... 

1
 k! ( 4 k
k 1
2
 1)
(1) k 1

k 1  2k 
 
k 



(  2) sin 2 1  1  log csc 1  log 2 sin 2 
sin( 2 k  2 )
k  1 k ( k  1)


 
=
.37252473265828081261... 

GR 1.444.1
1/ 3
 (1) 2 / 3  
1    1  
 1 




H 
H  1 / 3  H  1 / 3  

3    2  
2 
 2
 

1
( 1) k 1  ( 3k  1)



4
2k
k 1 2 k  k
k 1

 2k  2 
1
 k
1 .372583002030479219173...  24  16 2   
k  8 (k  1)
k 0 
1
.372720300666013047425... 
 coth    2 csch 2  i  (1) (1  i )  i  (1) (1  i )  

4


k ( k  1)
k 1
=  ( 1) k  ( 2 k )   ( 2 k  1)   2
2
k 1
k  2 ( k  1)


 
=
16 .372976195781925169357... 
.3731854421538599476...
 4 log 2 2 


x 2 dx
ex 1


1
 (k )
  k
  1 k
4
k 1 ( 4  1)k
k 1
1 .37328090766210649751... 
3
.373663116966915687102... 
GR 3.452.2
0
  
  1  i 
  (1  i )   1
  (1  i ) 

 cot 3 / 4   coth 3 / 4    
2  4   2

 2
 2


1
( 1) k 1  ( 4 k )
 
= 


4
2k
k 1 2 k  1
k 1


log 2
1
1
.37355072789142418039... 


 

3
6k  1
3 3
k 1 6k  4

( 1) k
= 
k  0 3k  2


1
dx
x dx
dx
=  3
  3
  2x
x 1
x 1
e  e x
1
0
0
.37350544572552464986... 

3
3/ 4
J80
K ex. 113

1 
1
1
2
 5 
 448 4  3 ( 3)    6 ( 3)    3 ( 3)    6 ( 3)    
186624 
6
3
3
 6 


 
=
.37382974072903473044... 
log 3 x
1 x 3  x 3 dx
 2k 

20 8 log 2


9
3
  k  4
k 0
k
1

(k  2)2


1
1 
 , Artin’s constant
p ( p  1) 
p prime 

log 3 k
( 3)
.37404368238615126287...   ( 3)   
k3
k 1
.37395581361920228805... 
3 .37404736674013853601... 
.37411695125501623006... 

4G 
2
4
 
=

0
4    2 log 2

6


 /2
  log 2 
1
x
2
0
x2
dx
1  cos x
GR 3.791.5
1

log 1  2  dx 

x 
log(1  x )
dx
x4
1
2

.374125298257498094399... 
( 1) k 1 k

k
k
k  1 2 ( 2  1)
.374166299392567482778... 
96  58 e 


1
 k ! 2 ( k  4)
k
k 0
.37424376965420048658... 
 (1) ( ) 

1
 (k   )
k 0

1 .37427001649629550601... 





1


2
2
k 2 k  k

k
k 2
i 1
(1  i )   (i )  i (i ) 
4
1 
 1
 coth     (i )   ( i )
 
=
2 4
2 4

 ( k  1)  1
.374308622850989741676...  
k2 1
k 1
 
=
.374487024111121494658... 
.37450901996253823880... 

1


3
2
k 0 k  k  k  1
2

e 1
2
cos1 log 2 
3
1
1
 
1
k
2

L
16
.374693449441410693607...  log
  kk
11
k 1 4 k


1
1   (k )
 (k )
 
 
.374751328450838658176...   k
2 k 1 k (2k  1)
k  2 2  (k )
k 1 2k  1

1
1/
.37480222743935863178...  e
1  
k
k  0 k! 
e1 / 
2
2 .374820823447451897569... 

7
1 .37480222743935863178... 

1
x2  2
2


2
0 x 4  3x 2  4 dx 
k 1 k  k  44




dx
dx
 x 2  x  2   2 x 2  x  1
1
i
i
.37490340686486970379...    1   2  e   2  e

2
 
=
 


 
=
 (1)
k 1
 

cos(k ) (k  1)  1
k 1
1 .374925956243310054338... 
1  J 0 (2 3) 

 (1)
k 1
k 1
3k

( k !) 2

 .375000000000000000000 =

3

8
e
k 0

=
 ( 2 k  1) log 3
1
 k ( 2 k  4)

k 1

=
1 .3750764222837193865...

log 3 x dx
1 x3
2 log 2
log 2 xdx
 
 log 2  
4 24
2
e4 x
0
1
.3751136755745319894...


 2 2
dx
2
2x
0
.37517287391604427859... 
 
=
  
 
1
Ei e  i  Ei ei   
2
1
  0,e  i   0,ei  
2
.375984929565308899765... 
4 2

105
.376059253341609274676... 
k

k 2

cos k
k 1 k!k

1  p 2

2
 p 4
p prime 1  p

1

k
1

1
 k
j 1 k  2
jk


.376148261725412056905... 

5 5
25  11 5 
 5 log


log

2
2
5 1  5 


1



k 1

k
 1
Li1/ 2     k
 4
k 1 4
 log 2
2 .3763277109303385627...  2G 
4
.37626599344847702381... 

3
5  20
 cot  tan

10
5
5

( 1) k 1 (2 k  1)!!
.3764528129191954316...  2 log 1  2  2 log 2  

(2 k )! k
k 1

(1) k 1  2k 
 
 
= 
4k k  k 
k 1
1 .376381920471173538207... 




coth  
6

5

1
2
 4k  5
k 1

65
1
.376700132275158021952...  9e 
 48  
e
k  0 ( 2k  1)!( k  3)
.376674047468581174134... 
k
2
Fk Fk 1

4k k
1
1
2
arctanh
1
arcsinh1 
2
2
2

( 1) k 1 ( 2 k )!!
= 

( 2 k  1)!!
k 1
.376774759859769486606... 

1

k
 2 (2k  1) 
k 1
k
1
1
2
1 .37692133029328224931...    G  log 2     ( x ) sin x dx 
2
0
.377964473009227227217... 
.37802461354736377417... 
1 .37802461354736377417... 
.378139567567342472088... 
.378483910334091972067... 
.378530017124161309882... 
.378550375764186642361... 

2

3
1
  3
2
24 16
k 1 k ( k  2)

1
1 1
1
.377540668798145435361... 
  tanh   (1) k  1 e k / 2 
2 2
4
e 1
k 1

1
1
1
 
B2 k  
 
=

2 k  1 ( 2 k  1)!  2 

2 2
1 .37767936423331146049... 
  ( 3)  4   (1) k k 2  ( k )   ( k  2 )
3
k 2
3
2

4k  k  2k  1
= 
k 3 ( k  1) 2
k 2

 log k 
.37788595863278193929...   1 

k2 
k 2 
.377294934867740533582... 

 (3)
GR 6.468


7
7
e(cos1  sin 1)  3

2
e(cos1  sin 1)  1

2
2  4 log
1

13
e
 log
e
 coslog x dx 
1
2
k 1
 4 k 
3
k 1 3 k
(1) k

k
k  0 2 ( k  2)

tanh
 cos1
2
x coslog x dx
1
 13
2



k
k 1
2 cos1  sin 1  si (1)  
1
3

4
2
1
 5k  3


sin x
dx
3
x
1


 ci(1) sin 1  si (1) cos1 
cos x
 (1  x)
0
2
dx 
J134
1 .378602612417434327112... 
 

.378605489832130816595... 
 
 ( 1)
k 1
k 2
.378696261703666022771... 
 
=
  Fk!Fk!


6
 (k ) log3 k

1
k
k 1
7776 (2) 
3

8

36 ( 3) (2)
4

k2
k 1
2
k
 ( k ) ( k )  1 
1295 (2) (2)



1
I 0 1  5  I 0 1  5  2 J 0 (2) 
5


1


1
 ...
2k
22

1 e
(1) k cosh k
e  e 1 / e   
.379094331700329445471... 

2
k!
k 0

H
3
2 1
2 .37919532168081267241...  2 log 2 
 2 log
  2 kk  2
2
2 1
k 1 2

k

3 
(1)
dx
.379347816325727458823... 

 log 2  
 3 6
2
x  x3
3
k 0 k  5 / 3
1
1 .378952026600018143708... 
k 1
.379349218647907817919... 
k
k 1
2
2
k
k 1
3 ( 3)
2
2
3
2
 log 2  6 log 2 
log 2 
 3  24 
2
4
2
1
=
  arcsin x log 3 x dx
0
.379600169272667639186... 

2

26


.379646336926468163793... 
1
 k (2k  1) log k .
k 1
.379651995080949652998... 
(1) k  (k )    1 
1 





2
1
J


 0 k 
(k!) 2
k 
k 2
k 1 


.379879650333370492326... 
3
2
1



32 40 20

.37988549304172247537... 
log


 
=
e
0
.37989752371128314827... 
3
2e

e 1
 /4

0
k
cos 6 x
dx
1  sin x
( 1) (1  2 k ) (1  k )


k!
j 1

dx

1
x
3


32 16
1
 x arcsin
0
3
x dx

.380770870193601660684... 
 (1)  (k )  1
k
2
k 2
  k(k (k1)) 


1 .380770870193601660684... 
k 2
e 1

2( e2  1)

2
.38079707797788244406... 
9 .3808315196468591091...

 
1 .3810359531144620680...
=

2
2 k 1
k 1
0
k 2
(22 k  1) B2 k
(2 k )!
88  2 22 

.38102928179240953333... 

 (1)k  2 (k )  1 
5 5 3 5  5 5 3 5 


 
 


2
10
2
10
2
5






k 1

(1) k Fk  (k )  1  

2
k 2
k  2 k ( k  k  1)
1 

tan
 5



1

2 2 log 2  7 (3)  
16

.381294821444817558571... 

.38171255654242344132... 

 (k )  (k )
k 2
2k

x sin x
 1 e
0

1 .381713265354352115152... 
x
 x log cos x dx
0
dx 
1
 ( k !!)
k 1
2
(1) k / 2 

k!
k 1
k 0

i 
i

k 1  (2 k )
.38189099837355872866...   log   1     1     ( 1)

 2  2
4k k
k 1

1 

 
=  log 1 


4k 2 
k 1

1 .381773290676036224053... 

 /2
.381966011250105151795... 
2 .381966011250105151795... 
cos1  sin 1   (1) k
2 
3 5

2
( 2k ) 2

(2k )!
( 1) k ( 2 k )!


2
k 1 ( k !) ( k  1)


7 5
1
1

 4   3  
2
 k 0 F2k

kH
5
5
1  log    k k
16 
4
k 1 5
k


(1)
1
  k
.382249890174222104596...   k
k 2 2  2
k 2 2  2
.382232359785690548677... 

GKP p. 302

2 .382380444637740041929... 

 ( 3k  1)
k!
k 1
.382425345822619425691... 
.38259785823210634567... 
.382626596031170342250... 


 ke
k 1


1
cot


2 2 5
5


 ( 2k )
k 1
2  log(1  2 )

6
 ( 3)


1
1/ k 3
5

k

 5k
k 1
1
2
1

arcsinh x
dx
x4
1



3
2 2
 sin  cos

4
8
8


1
1
k 1  ( k )
2
.382843018043786287416...   ( 1)
    2k
k
4 1
4
k 1
k 1 2
log 2 1 

 1  2i 
 1  2i 
.38317658318419503472... 
  
 
   (i )   ( i ) 
 2 
 2 

2
8
.38268343236508977172... 

=
.383180102968505756325... 
cos2 x
0 e x  1 dx
log

.383261210898144452739... 


 1

 (1) 
k 1
k 1
1
 k
k 1
2
k
( 2 k  1)  1 

1  log n k
.383473719274746544956...  
n
k!
n 1 n !2 k  2
log(1  x ) log x
x
0
1
.38349878283711983503... 
8G  2  4 log 2  16   

.38350690717842253363... 
 (2k ) (2k  1)  1
k 1
.383576096602374569919... 
.38383838383838383838
=
.383917497444851630134... 

H
4
4
log   kk
3
3
k 1 4
7

18

61
cot 2 2 

112 4 2

k
k 1
2
1
 6k  1
(k )  1 
2
1
.383946831127345788643... 

 
1  
k 
k
 (2)  (3)
k 1

H
.38396378122983515859...   3 k
k 2 k  1

2

1 .384161678...

 
2
( x)  1dx
2

2 .38423102903137172415... 
1

1  k  

2 
k 1 

.384255520734072782182... 

 1  k
k 1
.384373946213432915603... 
.3843772320348969041...
.384488876758268690206... 
1 .384569170237742574585... 
.384586577443430977920... 
.384615384615384615
=
=
.384654440535487702221... 

Q( k )

k
k 1 2

1 

 2
1
  log 2
  (sin 2 x) log ( x) dx
2
0
GR 6.443.1
1  i 
 i
 1 
  1 
 
  
 
  

 2
 2
 2 
8  2


 ( 4 k  1)
k
= 


k
4
4
k 1
k 1 4 k  1
1 .384458393024340358836... 
 


.384418702702027416264... 

2
2 2 k 1
 1

2 k 1
1
k 1 2

 
=
7 .38483656489729528833... 
.38494647276779467738... 
21
2
 12 log 
4
3

2

k 0
1
log(1  2 ) 
1
 3 (k  1)(k  3)
arctan x
x
0
k
1  x2
dx
1 2G   2 

 
 log 2 
3
3
6 12 6

2
2

GR 4.531.12
 /4

0
x2
dx 
cos4 x
log 3
log x
GR 4.232.3
 
dx
8
8
0 ( x  3)( x  1)
1
1
1
sin cos  2  log( 2  2 cos1)  (1   ) sin 2 
2
2
2

sin( k  1)
GR 1.444.1

k  1 k ( k  1)
5

13
log 2 2
log3 3
2
 2
 (3)  log 2 log 2 3 
log 3 
 Li2   log 
2
2
3
 3
 2
1
 Li3    Li3  
 3
 3
k 1


Hk
(1) H ( 2 ) k



k 2
2k k
k 1 3 k
k 1
 2
  ( 2)  
 3
 log 2  log x
  2
x 2
4 2
0

 /2


 





arcsin (sin x) / 2 sin x
Borwein-Devlin, p. 59
dx 
x
4

2
sin
0


7
1
1
 .3849484951257119020...  e   

3 k 1 k !( k  3)! k 1 ( k  1)!  2 k !

1
2/3  1
1/ 3
1/ 3
.384963873744095819376... ( 1)   4  2  i 2
3   (1)1/ 3  4  21/ 3  i 21/ 3 3
4

4
1
1 


2/ 3  1  2/ 3 

3 2
2 
2

 (3k )
.38499331087225178096... 
.385061651534227425527... 
2 .385098206176909244267... 
3 .385137501286537721688... 
.385151911060831655048... 
5 .38516480713450403125... 
.38533394536693817834... 
.38542564304175153356... 
.38594825471984105804... 


1


 


3

2
5
arctan

3
2

x
2
2
1
 x 1
3
13
6


 


1
1

  1 i 3  1 i 3   3
3 6
k  1 k  3k
29 

cos2 k
log 2 cos1   ( 1) k 1
k
k 1

i
i
i






 2  e  2  e
  (1) k 1 sin(k ) (k  1)  1
2
k 1




 1/
1
1
.385968416452652362535...  arccoth e   log tanh 
2
2


 
=
e
1
x
1 .38622692545275801365... 
(1) k 1 (4k  2)



2
k 1 ( k  1)!( 2 k  1)
e
k 0
2 k 1
1
( 2 k  1)
1 

2

 1
2 

 erf  k
k 1

e
x2
(1  x ) dx
0

.386294361119890618835... 

dx
 e x

1 .386075195716611750758... 

1  (1) k



3
3
4k
k 1
k 1 4k  1
k 2 k  2

1
1
J1 2   (1) k 1 k
2 k!(k  1)
2
k 0

=

1
2 log 2  1   k

k  1 2 ( k  1)

2
k 0
k
1

(2 k  4 )
J945
 

=
 (4k
k 1
2
1
 1) k
J374, K Ex. 110d
( 1) k 1

k  1 k ( k  1)

=

=
Lk
k
k
2
k 1

=

 ( k  1)  1
2
k 1

=

k



 (2 k  1)
k 1
4
k


 ( 1)
k 2
k
( k )
k
1  dx
 log1  x  x
1
1
=
J235, J616
2
dk
  E (k  )  1 k 
GR 6.150.2
0
1 .386294361119890618835... 
=

 
=
 
=
=
=
 
=
=

3k
 3
2 log 2  log 4  Li1     k
 4
k 1 4 k



1
1
Hk






k
k
2
k 1 2k  k
k  0 2 ( k  1)
k 1 2


 (k  1)
Lik (1)  Lik (1)  



2k
k 1
k 2

12 k 2  1

J398

2
2
k 1 k ( 4 k  1)

1
[Ramanujan] Berndt Ch. 2, 0.1
1  2 3
k  1 8k  2 k

1
Prud. 5.1.26.10

2
k  0 ( 2 k  1)( 2k  1)  1 / 4 

(2k  1)!!   2k  1
    k 

k 1 ( 2k )! k
k 1  k  4 k
1 n 1
x  x n  1/ 2  2 x 2 n  1
GR 3.272.1
dx
0
1 x

 
=
 
=
log(1  x )
dx 
x2
1

1


1
 log1  x  dx
0


 
=

1

 log1  x( x  2)  dx
0
3 .386294361119890618835... 

kH k
k
k 1 2

2  2 log 2 

k4

k
k  1 2 ( k  1)

k6
953 .386294361119890618835...  925  2 log 2   k

k  1 2 ( k  1)

k
.3863044024175697139...   k
k 1 4  1

1
28 .386294361119890618835... 
.38631860241332607652... 
27  2 log 2 
e
k 1
.38651064687313937401... 

4
k2
coth  

=
 ( 1)
k 1
k 1
2
4
csch 2 
1

 5

tan

4 4 5
2
.386852807234541586870...  log 6 2

1
 1
.386995424210199750135...  Li3     k 3
 e
k 1 3 k
.38716135743706271230... 
coth  csch 2 
2
1
2
k 2  ( 2 k )   ( 2 k  2 ) 
.386562656027658936145... 
.38699560053943557616... 
3

 4k
k 1
2
1
 4k  4
 /4
G
7
  (3)   2  x log tan x dx 
2
8
0
I 0 (1) I1 (1)


4
8
1
arctan 2 x
0 x dx

k3


2 k
k 1 ( k!) 4


3 .3877355319520023164...

1
 (k  1) log
k 2
.38779290180460559878... 
=
11 .38796388003128288749... 
.38805555247257460868... 
.38840969994784799075... 
2
k

5
5  5 5 log 5 
2
log


1

4
4
2
5 5
5


 (k  1)
1



2
5k
k 1 5k  k
k 1
2 2
 4 (3) 
3


k!k ! 
k 1



x
2
Li2 (  x ) 2 dx
0
 1  (2k )!


1
arcsinh 2 1
log 2 1  2 
2
2

 5k
k 1
1
k
2



=
 (1)
k
k 0
(2k )!!
1

(2k  1)!! 2k  2
J143
1
=
arcsin x
dx
1  x2
0

1
=

arcsinh x
0
.388730126323020031392... 
dx
1  x2
34

15
CFG D1
2k
1  2 k ( k  1)


 k!
3 k 0
k 0 k!

7 .389056098930650227230... 
961 .389193575304437030219... 
.3896467926915307996...

e2  1 
6
12 log 2  2 

.39027434400620220856... 

2
6

1 .39088575503593145117... 
.3910062461378671068...

k 1
4
k


 4k
k 1
3
1

 k2


 k!3
k 0
2 k 1
1
(2k  1)
S 2 ( 2k , k )
 S ( 2k , k ) 
k 0
2 .39074611467649675415... 
 ( k  2)

1

1
2
.39053390439339657395...  erfi 
3


1
2 .39064...
  2
k 2  (k )



 1  k (3k  1) 
k 1
.3906934607512251312...
LY 6.40
1
8 log 2  2  2 

 (k  3)
k 0
2k


 2
log       
 3 

6 
2
1
  arcsin x arccos2 x dx
4
0

sin 2 k
e(e  1)(e  1) 2


4(e  1)(e 2i  e)(e 2i 1  1)
ek
k 1


1
.391235129453969820659... 
tanh    2
8
k 1 4k  4k  5
2i
.391031136773827639675... 
2 .39137103861534867486... 
.391490159033032044663... 
 2 log 2
3
log 3 2


3
1
Lie   
e

1
e k
k 1
k
e


log 2 x
0 ( x  1)( x  2)
GR 1.422.2, J949
.39173373121460048563... 

.39177...


sinh 1

3

1
1
1 k
s2 k 2
s
 (k )

.391801000227670935574... 
 1  dx
3 4 
x
 cosh x
 k (k  1)

k 2
.391892330690243774136... 
2

2
6

  (k )



 (1)   (2k )  1
k
k 2
 4 log 2  log 2 2 

k2

k
2
k 1 2 ( k  1)
2
  cos x

x cos x
.39190124777049688700... 
dx   
dx  
sech

4
2 0 sinh x
 0 cosh x 
( 2)  1
6
.39207289814597337134...  1  2 

( 2)
2
1 .392081999207926961321... 
1 .392096030269083389575...
5 .392103950584448229216...
2
 3/ 2
4



 1  5 
1  1 5 
  H

  H 
 2  
2   2 


3
3
 

 F  (2k  1)  1
k 2
2 k 1

.39225871325093557127... 
e
2
   e1 x x log x dx 
4
0
.3925986596400406773...
2    erfi(1)   Ei(1) 
=

1
 k ! k (2 k  1)
k 1

218 .39260013257695631244... 
.392699081698724154808... 
4k k
4e 4  

k 1 k!

8

=

  4(k1) 2
 arctan 2  1 

k
k 0
1
 (4k  1)(4k  3)
k 0

=
( 1) k 1
2k  1
sin

2
2
k 1 ( 2 k  1)

( 4 k  1) ( 2 k )
1




k
2
16
k 1
k 1 4 k  1

 2k 
1
 

k
k  0 4 ( 2 k  1)( 2 k  3)  k 

=
=

J523

1
 16k
k 1
2
1
  k  7   k  6   k  1  k  4  

 Plouffe



k   8   8   8   8  
k 1



1
arctan
[Ramanujan] Berndt Ch. 2

2
 (1  k 2 ) 
k 1


 
=
=


 2

 
=
1
( k 1) / 2 

1
  ( 4 s  2)
r 1 s 1

=
x
2
0
e
2x
0
J1122

1
dx
0 x 4  4 

=

dx
 16

=
2r
xdx
dx
0 x 4  4  1 x3  x 1 

 (x
0
x dx
 1) 2
4
dx
 e 2 x

=
sin 2 x sin 2 x
dx
0
x

=
sin x 2  x 2 cos x 2
dx
0
x5
1
=
 x arcsin x dx
GR 4.523.2
0



=
  x e  x log x cos x dx 
0
4 .39283843336600648478... 
2 .39292255287307281102... 
8
2160
8G 
 4 (4) L(4) 
2
2
arcsin 2 x
 x2
1
(1) 2 2k k 2

(k!) 2
k 1

 2 J 0 (2 2 ) 
.393327464565010863238... 
7 (3)  2 log 2


4
4
.393469340287366576396... 
2 .39365368240859606764... 
r (k )
4
k 1 k
1

.393096190540936400082... 
1 .39340203780290253025... 



Hk
 (2k  1)
k 1
32 15

11(4  15 ) ( 15  3)
e 1

e
6
2
k 1
(1)
k
k 1 k ! 2


2


k 1

1

 1 k (k  8) 


.39365609958233181993... 
1
 k!k
2  (2 k )  1

k
k 3

1 .39365735212511301102... 

k
2
k 2

 2
2 

 Li  j
k 2
2



k 
(1) 

1 .393825208413407109555...   1 

 2k  
k 1
 k  
 k 

.393829290521217143801...  4  3 (3)

18 .393972058572116079776... 
.39416851186714529353... 
4 .39444915467243876558... 
50

e

(1) k 8k

k!
k 0

coth 
8
4 log 3
(1) k 1 H ( 3) k

2k k
k 1

3
k
 (2k )  1 
log    
4
k 1 k  1

.39463025213978264327... 
.39472988584940017414... 
.394784176043574344753... 
2 .394833099273404716523... 
.394934066848226436472... 
=
2
25
1
I1 2 2 
2
   k!(k2  1)!

2
k
k 0
5
  (1) 3   (2,3) 
6
4

1
1
 Li2 (e)  Li2    i   1
2
e



=

 ( 1)
k

1
k
k 3
2
( k  1)( ( k )  1)
k 2
 
=
(1) k 1 k

j k 1
j  2 k 1

=
log x
dx
4
 x3
x
1
1
=

0

1 .39510551694656703221... 
x 2 log x
dx
1 x
  (k )
k 3
Berndt 5.8.4

.395338567367445566052... 

1
 1  k !
k 2
.39535201510645914871... 

 (1  e)

e(1   2 )
e
 x /
cos x dx
0
 5
( 1)  7 
1 .39544221511549034512...         
 8
 8
( 1)
.395599547802009644147... 

( 3k  1)( k  1)
 (k  2)

8k
k 1

k
 k!(k  5)
120  44e 
k 1
1 .395612425086089528628... 
e1 / 3 

1
 k!3
k
k 0

1 .395872562271002...


1
  1 k
n2 k 2
2
.395896037098993835211... 
3e
19


8
8
n

2k k 2

k 1 ( k  3)!

1 .3959158283010590...

 1  1/(2k )
2
1
k 2
51664  2

 H ( 2) 7 / 2
1 .396208963809216061296... 
11025
3
2

k 1  ( k  1)  1
.39660156742592607639...   ( 1)
k!
k 1

1
k2
1 .39689069308728222107...   2 log
k
k 1 k
.39691199727321671968... 
2 sin 2  1 

 (1)
k 1
k 1
2k k

( 2 k )!( k  1)

1

2
1 

csc
sin 
  1  2  
.39710172090497736873... 
3 k 1 
3k  1
2
3

k
.397116771379659432792...   2
2
k  1 ( k  1)
1
.397117694981577169693...  erf  
e

(1) k 1 4k
1 .397149809863847372287...  1  J 0 ( 4)  
(k!) 2
k 1
.39728466206792444388... 
3 3
4 2
.397532220692825881258... 
2 cos1

e
sin 3  x 2 
dx
  
x2
0


x 2 sin x
1 e x dx
J1064
Prud. 5.1.25.29
.397715726853315103139... 
1 log 2


6
3

1
 (2k  1)(2k  4) 
k 0

5e
k2
 
4
k 1 ( 2k  2)!

|  ( k )|( 1) k 1
.3980506524443333859...  
2k
k 1
3 8
7116 .39826205293050534643... 
4
3 .397852285573806544200... 
1
.3983926103133159445...

 arctan(arctan x) dx
0

2 1
.39890683205968325214... 
coth 
 
3 4
2 6
1
.39894228040143267794... 
2
 ( 3)
.398971548420202857300...  1 

2
.3990209885941838469...

.39920527550843900193... 
.39931600618316563920... 
.39957205218788780748... 
 
=
k 1
1
2
2

sin 2 x
cos 2  cos 2 si (2)  sin(2) ci (2)  
2
( x  1) 2
0

1

 17
1

tan
  2
4
2
17
k  1 k  5k  2
2
k

e 1
2
 
16
k  0 k !( k  2 )( k  4 )
2 arccsch 2 2 
 (1)
k 1

 3k




k 1
  3  5 
3 5 
1 
 
  Li2 
2 log 5 (arccsch 2)  Li2 



2
2
5 




Hk

 2k 
 k
k
.4000000000000000000
=
=
=
2

5

 (1)
k 1
k 1
1

2 cosh log 2
1  p 2

2
p prime 1  p

=
sin 2 x
0 e x dx 

Fk

2k
 (1)
k 1

 (1)
k 1
kFk
2k
e(  log 2 )( 2 k 1)
k
J943
k 0

sin 2 x
0 e x dx 


0
cos x
dx
e2 x

2 .400000000000000000000 =
12

5
F2 k k

k
k 1 4
1 .40000426223653766564... 
  (k  1)  1H

k 1
2 k 1
.40009541070153168409... 
 log 2
.400130076223970451846... 
e G
.40037967700464134050... 
e    1  Ei (1) 

k

2 
k  1 k !( k  1)

1
 k !( k  2)
k 0
2

1
=
  ex x log x dx
0
 (k )

.400450020151755157328... 
 ( 1) k
2k  1
k 2

.400587476159228733968... 


1
  k

k 1
1
1 
arctan

k
k
  (k ) log  (2k ) 
k 1
.400685634386531428467... 
.400939666449397060221... 
 ( 3)
3
cos( k / 3)
k3
k 1



 
1 
4 2

cos 2  sin 2
cosh 2  sinh 2  
4 2





 
=
.40130634325064638166... 
sin 2 x dx
0 1  x 4
1 
 2 ( 1)
6

 (k )
( k  1) ( k )

k!
k 2
e

 e1/ k  1
k

k 1

 ( k  1)( k  2)
k 2
Adamchick-Srivastava 2.22

1 .40147179615651142485... 
.40148051389327864275... 

log 2 

 
1/ k

7
1
 2

24 k 1 8k  44k  48
(1) k 1 k

2
k 3 k  4

(1) k


k
k  0 k !3 (k  2)

.40162427311452899489... 
12
9  1/ 3 
e
.40175565098878960367... 
5 2
3


12
16
1 .40176256381563326007... 
0
 4
 3
dx
5
6
   1   
1

0
1 .402203300817018964126... 

cos5 x
dx
1  sin x

 1 
log      
 3 

1 .402182105325454261175... 
=
 /4
1 x



3
0
3 2
6 
4
1  1
   
6   3  6 
1
x dx
Seaborn p. 168
1  x6
 /2
 arccos
0
0
3 2
7 .402203300817018964126... 
4
3  2
.40231643935578326982...   
csc  2 
4
4
.40268396295210902116... 


=
.402892520174484499083... 
1
 x 3 sin x dx 
3
x dx
(1) k


2
k 2 k  2

4 (3) 4
2
  (2) log   log 3  
5
5
3
2
 5  1  4 (3)
5  1
2
 


log
 2  5
2 
15


Li3   2  
2 2
log 
3

5 1
2
 
1 i 2
 1 i 2 
1  i 
i 

 
  
 
  
   
 2 2
4 2 2
4 
4  
2
2
(1) k  1 k

2
k 1 k  1 / 2


 
=


1   1  5 
2 
 
  Ei  
Ei 
arc
h


2
csc
2




2
5 
 1 5 



k!
3 .40301920828833358675...   k  1
k 1 k

 (k )  1
.403025476056584458847...  

k !!
k 2
1 .4029920383519025537...

.40306652538538174458... 
2

9 3

1


 2k 
k 1
15 
k 

1


2
27k
k 1 27 k

3
2
2

x
0

k 1
x dx
 27
3
Fk
 k! k
1 .40308551457518902803... 
1/ 18
e
6 .40312423743284868649... 
1 .403128506816742370427... 
1 .403176122350058218797... 


1 
1 
 
erf
2
3 2


1
 k !!3
k
k 0
41
 2 
 1
 
0 F1  ;1;   I 0 
 e
 e

 2 (2k )
k 1
( 2 k )!


1
 (k !)
k 0
2
ek
1
2 .40324618157446171142... 
1 3  1
log x
dx 
    
3
8  3
x (1  x 2 ) 2
0
GR 4.244.1
6

98
k 1 k
.403292181069958847737... 
  (1)
243
2k
k 1

(1) k 1
.4034184004471487...
  3
3
k 1 k  k
.403520526654739372535... 
K0 (2)  log 2

2


0
cos2 x
1  x2
dx

dx
.403652637676805925659...  1  e Ei ( 1)   x

2
0 e ( x  1)

x dx
 e ( x  1)
x
0

1 .403652637676805925659... 
x 3dx

2  eEi (1)   x
e ( x  1)
0

.403936827882178320576... 
2
k 1
1
2
k
1

1
(1) k 1



k
k
k 1 3  1
k 1 3  1
(2)
 2 ( 3)  2    ( 2 ) 

.404063267280861808044... 
.4041138063191885708...

2 .4041138063191885708...
1
=
log 2 x
1 x 3  x 2 dx 
x log 2 x
0 1  x dx 

2 (3)    ( 2 ) (1) 

k 1
(1) k 1


k 1  2 k  3
2   k
k 


 
=

 
=

 (1) (k )
k 1
k

Hk
k
2


x2
0 ex (ex  1)
GR 4.26.12

log 2 (1  x )
dx 
= 
x
0
1

 
1

 
=
x 2 dx
0 e x  1 


1
log 2 x
dx 
x2  x
2
log x
 1 x

0
1 1

 
3 .4041138063191885708...
log( xy )
dx dy
0 0 1  xy
=
 

2 (3)  1 
=
 ((2 k  1)  (2k )   (2 k  1)
9k 4  2k 2  1

k ( k 2  1) 3
k 2


2
k 1
15 .40411960755222466832... 
1 .404129680587576209662... 

8G   
(3k  1)( k  1)
 
 ( k  1)
4k
k 1

2

1
coth 3  
2
6
.404305930665552284361... 

2



sin 3x
dx 
x
1
0
e





2 
190  190 47  77 5 arc csc 2 1  5  190  47  77 5 arc csc h 2  11  5  

1805 

k
1

(1) F k

 
= 
 2k 
k 1
 
k 
2 2 k 1  (2 k )  1
1 .404362229731386861519...  

(2 k  1)!
k 1

k  (k )  1
.4045416129644767448...   ( 1)
 (k  1)
k 2

 csc 
.40455692812372438043... 
2


1
2
 k sinh k
k 2
1  ( 1) k 1

2 k 1 k 2   1

2 .404668471503119219718... 
.404681828751309385178... 
.404729481218780527329... 
k2

k
k 1 k
 1
1    Ei  
 e
9  18   2 

1
 k !e k
k 1

k

i
Li  e 6i   Li3 ( e6i ) 
2 3

=
.40480806194457133626... 
sin 6k
k3
k 1


2
coth    
GR 1.443.5
1
1   1  i    1  i   
2

k 1
=  3
  (1) k 1  (2k )   (2k  1) 
k 1 k  k
k 1


 
.404942342299990092492... 
cosh 2 sinh 2


2
2 2
.40496447288461983657... 
  (2)
.405182276330542237261... 
2k k


k  1 ( 2 k  1)!
cos(sin 2 ) cosh(cos 2 )  sinh(cos 2 ) 


cos 2 k
k!
k 0

k2
(1)

k5  1
k 1

3 cos1
(1) k 1 k 2
.405226729401104788051... 
 
4
k  1 ( 2 k  1)!
 /2
e
2 .4052386904826758277... 

2
.4051919148243804...


k 1

.40528473456935108578... 
4
2

3
1
 1
.405465108108164381978...  log  Li1     k
 3
2
k 1 3 k
k 1


( 1)
( 1) k
= 
  k
k
k 1 2 k
k  0 2 ( 2 k  2)

1
1
= 2arctanh  2  2 k 1
(2 k  1)
5
k 0 5

=
.405577867597361189695... 
dx
1 ( x  1)( x  2) 


2 15

.40573159035977926877... 

H

0
dx

x
e 1
K148

0
dx
2
5
 2 (k  1)

k
k
k 1
 (k  2)

e1/ 2 k
2
k !2k
k 0
k 1 k


1
1
k 1  (2k )
1 .405869298287780911255...   ( 1)
  arctan 
k
2k  1
k 1
k 1 k
2 .40579095178568412446... 
.40587121264167682182... 
1 .406166544295979230078... 


4
240



1


log 3 x
dx
x3  x
 ( 2 ) ( 3) 
dx
x
1
 2e
 3x
0
( 3)
log 3
J102, J117
1 .4063601829858151518...


 (3k )
k 1
k3

7 (3)
.40638974856794906619...  2    2 G 

2
2 .40654186013...

.406612178247656085127... 
.40671510196175469192... 
1 .40671510196175469192... 
 

 
120 (5)

5

 e

k
b

0
x 2 arccos2 x
dx
1  x2
1
 ka
dx
x1 / 5
0
1
1
sinh 2 1
 
4
2
 sinh
sinh 2 1
 
4
2
1
2
x dx
0
 cosh
2
x dx
0

1
1
3arccsch 2
arccsch2
5 5
5 5




log

5 2 2 5 10 (1  5 ) 2 5 (1  5 ) 20(1  5
5 5
1

 
5 log 2  log( 5  1) 
5(1  5 )
1 7
 1 
       
=
10   10 
 5 
.40690163428942536808... 

sum of row sums in
1

(1) k

k  0 5k  2

 (2k )
2 .40744655479032851471...  

k!
k 1


=
1 .40812520284268745491... 

e
1/ k 2

1
k 1
1 2

 log 2 
6 18
1
log 2 x
0 ( x  1) 4 dx
6

6

1 

.40825396283062856150...    1  k  
2 k
k 1 

49
H6
1
.408333333333333333333 =

  2

120
6
k  1 k  6k
.40824829046386301637... 

k
k 4
2
1
9
2   sin(2  )  sinh(23/ 4  ) 
1 
 4 
 
.40837751064739508288... 
16 
cosh(2 3/ 4  )  cos(23 / 4  ) 
7/4
k 1
(1) k
4
k 1 k  1
.408590857704773823199...  14  5e

.408494655426498515867... 

3/ 4

k
k 1
4
1
2
.40864369354821208259... 
sinh 1


erfi1  erf 1 
2
8

 1  dx
2 4 
x
 cosh x
1
Bk
1 
1 
Li2   
6
e
k 1 ( k  1)!k

1
=  k 2
k 1 e k

2
.408754287348896269033... 


2 .40901454734936102856... 
97 .409091034002437236440... 
[Ramanujan] Berndt Ch. 9

e 

2
e
1 x 2
dx
0
 1
 2
 4   ( 3)    90 ( 4 ) 
(k  1)(k  2)(k  3) (k )
2k  4
k 4
1
4
( 3) 
193 .409091034002437236440...    96      
 2

dx
9 (3)
5 .4092560642181742843... 
  x1 / 3
2
1
0 e


 
=
.40933067363147861703... 
18 .40933386237629256843... 
1 .409376212740900917067... 

e(sin 1  cos1)

2
 2  e
3
22
 log
x sin log x dx
1

Borwein-Devlin p. 35
 (1)  (2k )  1
k 1
k 1
2

3


.409447924890760405753... 
e
2

a (k )
3
7
k 1 k

 (k )  1
.410040836946731018563...  

k  2  (k )  1
1 .409943485869908374119... 


Titchmarsh 1.2.13

.41019663926527455488... 
H ( 4) k

k
k 1 3 k

1 .410278797207865891794... 
2
 Fk
k 1

2 .410491857766297...

 (1)
k 1
k 1
e1/ k
k2
1
tan 1
arctan

.41059246053628130507...  1 
2
2
1
cos2 x
0 1  cos2 x dx 
(1) k 1
 
k
k 1 k!!2

.4106780663853243866...



k!2k
1
1
2k
e
erf



1 .41068613464244799769... 

2
2 k  0 (2k  1)!! k  0 (2k  1)! k 1  2k 
k! 
k 


e
k!2k
(2k )!!
 1 
erf 
2 .410686134642447997691...  1 
 
 
2
 2  k  0 (2k )! k  0 (2k )!
.410781290502908695476...  sin e
.410861347933971653047... 
.41096126403879067973... 
1 .41100762619286173623... 
.411233516712056609118... 

3
(1) k
3  9 log   k
4
k  0 3 (k  2)

 ( k ) ( k  1)  1
(1) k

k
k 2

1

k  1 k !! k
2
 ( 2)

  Li2 (i )  Li2 (i ) 
24
4

=
e
x2
k 1
2
1

1
log x
x log x
1 x 3  x dx   0 1  x 2 dx

=
1
 4k
x 3dx
0
=

 log1  e  dx
2 x
0



1  dx

2
 x
 log1  x
0
 /2
=

 log(sin x ) tan x dx
GR 4.384.12
0
.41131386544700707263... 
1
 cos1 cosh 1  sin 1 sinh 1  1 
2
1 .41135828844117328698... 
 cos x sinh x dx
0

3  3
(1)

csch
  2

2
2
3
k 0 k  1/ 3

1
SinhIntegral (2)


1

k  0 k!( k  2 )! ( 2k  1)

2 .411354096688153078889... 
1

 1
k
8  8  8  8... [Ramanujan] Berndt Ch. 22
2 .411474127809772838513... 
2 3 cos
1 .41164138370300128346... 
2  1
 sinh (log  )   i sin (i log  )
2
6 .41175915924053310538... 
7G
18
.4117647058823529
=
.411829422582171159052... 
7
17
 sech


 
=
k
k 1
.41197960825054113427... 
 3
2
3i 2  3i 2 
 
 
 
 

 2   2 

k 1
k k
 k 1
2
2

log( x  2)
dx
x4
1
3 log 3

8


1 .41228292743739191461... 
1 .412553231710159198241... 
csc 1 
2
2

1

.4126290491156953842...

log x dx

x  1
0

csch  2 sinh 2  
  2 
k 2
k 4  2k 2  1


k 4  2k 2
k 1

 ( k  1)  1

 (k )  1 

5 .4127967952491337838...


k2


k
k 1 2  1

.412913931714479734856... 
k
k 2
1
log 3 k
 (2k )

.413114513188945394253... 
3
 ((2k )!)
k 1
148 .413159102576603421116... 


 ( 4 k  1)
4k  1
k 1
.41321800123301788416... 
=
=

k
k 1
.413292116101594336627... 

3
1
3
cos(log  )  Re i 
  

1
  1  n   2 

k 
n  2  k 1 


2 .4134342304287...


1
6  3 log 2  3
9
2 2 3
1

arcsinh
3
9
2

( 1) k 2 k
 1 1
, , ,   
2 F1  11
 2 2
k 1  2 k 
 
k

.413261833853064087253... 
1  2 
 2  
  1
  J0 
0
 k 
k
  2 I 

e5 

.41318393563314491738... 
2

 1  k
1 

 1
3
.413540437991733040494... 
2

3 3  52 / 3

x
0
dx
5
3


 2k
  (4k )!
1 .41365142430970187175... 
1
cos   cosh  
2
6 .41376731088081891269... 
16

243
2 .414043326710635964496... 
2  2e  2  erfi1 

4
k 0
3
log x dx
x3  1
0


1
 k!(k  1/ 2)(k  1)
k 0


2 .414069263277926900573... 
.414151108298000051705... 
.4142135623730950488...

1 .414213562373095048802... 
e 1
   e x sin 2 x cos x dx
10
0
log12

6
H2 k 1
k
k 1 4

(2k  1)!!
5


2 1  
 tan   cot
k
8
8
k 1 ( 2k )! 2

2

=

1  2k 

k 
k
8
k 0

(1) k 
=  1 

2 k  1
k 1 

3
2 .4142135623730950488...  1  2  sin
csc
8
8
 3  i 3
1  log 2
3  i  1 i 3
 
 
 

.414301138050208113547... 
3
6( 3  i )   4 
 4  


  3  i 3
 1  i 3 
2i
 
  

6( 3  i )   4 
 4  
(1) k 1
=  3
k 1 k  1

1
.41432204432182039187...  
k  1  3k 
 
k

.414398322117159997798... 
2 .4143983221171599978...


1
 3

7 (3)  8    3,   
3
 2
k 1 ( k  1 / 2)

k
7 (3)  6   2
1 3
k 1 ( k  4 )
GR 0.261
8 .4143983221171599978...

7 ( 3) 

1
 (k 
k 1
.414502279318448205311... 
=
.414610456448109194212... 
.4146234176385755681...

=
1
2
)3
1 1
  E (i ,1)   (1  i )   (1  i )   (1  i ,1) 
e 2

( 1) k 1

2
k  1 k !( k  1)
1
7

3

cot 

6 2 6
2

 2k
k 2
1
2
3

4
10
2


3
9
9 3

k

k 1  2 k 
  ( k  1)
k
.414682509851111660248... 
1
1  (k )
  k
p
2
2
p prime 2
.41509273131087834417... 
log 2  2   


 k  2  (k  1)  1
k
k 1
 (k )

=
2  (1) k
k 2
.415107497420594703340... 
.41517259238542094625... 
2
k (k  1)

e 

H ( 3) k

k
k 1 3 k

1 .415307969994215261467... 
  o (k  1) (k )  1 
k 2
.415494825058985327959... 
=
27 .41556778080377394121... 

25 2

9
   ( jk )  1)  1
j 1 k 1
10

  4  (4 k )  1 
k
k 1

log x dx
x6/5  1
0

 k 3  (3k )  1 
k 1
31 .415926535897932384626... 

11


coth  2  cot  2 
6 2 2

4

4
k 2 k  4

.41562626476635340361... 

k 4 (k 6  4k 3  1)

(k 3  1) 4
k 2

.415939950581392605845... 
1 .41596559417721901505... 
 (2)  1
G
2
 1
2
4

3
36


f 2 (k )

k2

k 2


1
  ( jk )
j 1 k 1
2
(1) k 1 k 3


2
2
k 1 ( k  1 / 4)

1

2
1

 
=
 E ( x )dx
2
Adamchik (17)
0
.416146836547142386998... 
1 .416146836547142386998... 
(1) k 1 4k


(2k )!
k 0

 cos 2 
2 sin 2 1  1  cos 2 

 ( 1) k 1
k 1
7 .4161984870956629487...

7 .416298709205487673735... 
.41634859079941814194... 
4k

( 2 k )!
GR 1.412.1
55
1 1
 4 4
arctan 3

3
 ,  
1
2  
  4

dx
0 x 2  2 x  10 
1

3
dx
.41652027545234683566... 

2
16 2 0 (2 x  1) 3
.416658739762145975784... 
 ( 3) 

4


.416666666666666666666 =
5

12
1
 2k
k 2

 (2 k  1)
.41705205719676099877...   

4k  1
k 1

|  ( k )|
.41715698381931361266...   k
k 1 4  1
.41716921467172054098... 
7 (3)  2  3
32


 4G 

2
2
8
27
.417418352315624897763... 
3 
e
erfi 1  
8
4
.417479344942600488698... 
  ( k  4) 
 Im
 
k
 k 1 (2i ) 
k
2


k
 k !(2k  3) 
k0


 4k
k 1

.41752278908800767414... 
k ( k  1)( ( k )  1)

4 k 1
k 2

5
2
 k3
1

k 1 k ( 2k  1)
 (2)  4 log 2  1  
2
 (1)
k 1
 (k  2)
2k
.417771379105166750403... 
1 

3 2

sin x 2  x 2 cos x 2
dx
0
x4


log 2
(1) k

 

6
6 3
k  0 6k  2
1 .417845935787357293148...  root of  ( x )  3
.41782442413236052667... 
1
e 2
.418023293130673575615...  1 


e1
1 e

B
=   k 
k!

2

2
k 1
log 2 
3
  (2) 
4
.418155449141321676689... 
Im (i )  Im (i )
1
3
dx
 x 3
1
k
1  e 
1/ 2 k

log( 2 k  1)
2 
k  0 ( 2 k  1)
GR 1.513.5


Prud. 5.5.1.5
 (i )
.418168472177128190490... 
2 .418399152312290467459... 
x
k

.41835932862021769327... 

1
 e  1



k 1  e  k ( k  1)

.418115838076169625908... 
24
GR 3.852.5
1
93 (5)  7 2  
32
4

3 3
1
arcsin 2 x arccos2 x
dx
0
x

k ! 3k

k
k  1 2 ( 2 k  1)!!
=
area of unit equilateral triangle
=
(k  12 )!(k  12 )!

(2k  1)!
k 1
J278
CFG G13


 


=

0
dx

2
x  x 1

=
e

.41868043356886331255... 
.41893853320467274178... 
1 .419290313672675237953... 
x

dx
 1 x
3/ 2
0
dx

 e x  1
35 3

2592
sin(5k / 6)
k3
k 1



GR 1.443.5
2k k
 (k  1)  1

k 1 k  2
2
2
 1  i 3 
i   1  i 3 

 3
  
 

tanh

2
3
2 3   2 
 2  
1
log 2  1 
2

 1 i 2 
i  ( 1)  1  i 3 
 
   (1 ) 

2 3
 2 
 2  

=
97 .41935701625712157163... 
k
k 1
4
Hk
 k 1
   4 

.41942244179510759771... 
2

 1  2
k 1
2 .419550664714475510996... 
k
1 

 1
4 log 2  4 
2
6
 2 (3)  8 

k3
 (k  1)  1

k 1 k  2

e sin 1
sin k
.41956978951241555130... 
  k
2
1  e  2 e cos1
k 1 e
2
2 .4195961186788831399...  sec(log  )  i
   i

( 1) k 1 k
1
1
2
1
.4197424917352996473... 
cos

sin
 

k
4
4
2
2
k 1 ( 2 k  1)! 2

148 .41989704957568888821... 

e5  e5
.42026373260709425411... 
7

2 2
 4   (2k )  1
3
k 2
2 .42026373260709425411... 
9 
2 2

3
1 .420308303489193353248... 
 2 (3)

 (6)

=
3 .42054423192855827242... 
 ( 1) k 1
( 1) k 1 




2
( k  1 / 3) 2 
k  1  ( k  1 / 3)

2 ( k )

3
k 1 k

HW Thm. 301
 1  p 3 
  3 
p prime  1  p


2
 log 2
   x log sin x dx
2
0
GR 4.322.1

5
1
 3 log 2   k
2
k  0 2 ( k  1)( k  3)

k 1 k !
.420571993492055286748...   ( 1)
k k 1
k 1

1
.420659594076960499497...   3
k 2 k  5
.420558458320164071748... 

17 .420688722428817044006... 
8 log 2 

0
ex x 2
 ex  1 3
dx
GR 3.455.1

.42073549240394825333... 
sin 1  ( 1) k 1 k

2
k 1 ( 2 k )!
.421024438240708333336... 
K 0 (1) 
.421052631578947368
=

0
0
cos x
 sin(tan x) sin x dx  
dx
1  x2
8
19

.421097686033423777296... 
 /2
1


k
k 1 4  1

 0 (k )
k 1
4k



4k  1
 4 4
k 1
k2
k

1


e
 e
   
  e k 1   
k
6 .42147960099874507241... 
 (1) k 
1  k 2  

2 k 
k 1 

StirlingS 2(k , 2)


k 2 StirlingS1( k , 2)

 (k )
(1) k 1 k0

2 1
k 1

(1) k
2 J2 ( 2 )  
k
k  0 k !( k  2 )! 2

1 .4215341675430081940...

.4215955617044106892...

.421643058395846980882... 
.421704954651047288875... 

.421751358464106262716... 
3 log 2 9 log 3


2
16
.42176154823190614057... 
 e
 
=
1 .421780512116606756513... 
.421886595819780655446... 
.42191...

.4219127175822412287...

1 .4220286795392755555...


sin 6 x
0 x 3
GR 3.827.13
sin(2 sin 2)  sin(2 sin 2)cosh(2 cos 2)  sinh(2 cos 2) 


 2i
2 k sin 2 k
i 2e 2i
 
e  e2e
2
k!
k 1
2

k Hk

k
k  1 2 ( 2 k  1)
sin5 1

1 n

2   (k )
n 1 n k 1
2

Hk

3
k 1 ( k  1)
log 3 2  2 2 2 log 2
 1


 4 Li3     3 (3) 
 2
3
6
3



2 cos 2

x log 2 x
 
= 
dx
( x  2)( x  1) 2
1
 ( 3)
.422220132000029567772... 
 ( 2 )   ( 3)
.422371622722581534146... 
5 .422630642492632281056... 
.422645425094160918302... 
.422649730810374235491... 
.42278433509846713939... 
=
2
12

 (1  k )
k 1
2k k


3 1
( 1) k
 
  3 log
3  1 k  0 ( k  1 / 6)
2 1 2
 log 2  1   2 2  1 
16 4

1
(1) k 1  2 k 
1
 
 
2k  2 
3
k 1

 (k )  1
1    (2)  
k
k 2


 1 1
k
1

 Li1        log
 

k 1 k 
 k  k  k 2 
k 2 


=
  log 2 

0



Berndt Ch. 9
K Ex. 124g
x log x
dx
ex

=
1  dx
 sin x


x
1 x  x
0
 


 
=

0
1 .42278433509846713939... 

1  dx
 cos x 

1 e x  x


2
2  
k

 (k )  1
k
k 2

1 .42281633036073335575... 
GR 3.781.1
k 

2k

 1  2
k 1
.422877820191443979792... 
 2
Li3  
5
.422980828774864995699... 
CosIntegral (2)    log 2  
(1) k 4k
k 1 ( 2k )!( 2k )


.423035525761313159742... 
  (2k )  1
2
k 1
(1) k


k
k  0 2 (3k  2)

4
 (k )
.4232652661789581641...  (3) 
  3
360
k 1 k
.423310825130748003102...    e
.423057840790268613217... 
1 .423495485003910360936... 
1  2 5 1
2 F1 1, , ,  
2  3 3 2
 (2)   (3)
2

AS 5.2.16
.423536677851936327615... 
log 2 
1  1 i 
i
i 
1 i 


 
  
   1     1    
4  2 
 2 
 2
 2 
(1) k 1

3
k 1 k  k

=
.423565396298333635...

=
1
1
1
2  log(e  1)  2 Li2    2 Li3    Li2 (1  e)  2 (3) 
2
e
e
k


1
(1) Bk
Bk
  

6 k  0 k!
k  0 ( k  2)!
H k  (k  1)  1
2k
k 1

.423580847909971087752... 
2 .423641733185364535425... 
.423688222292458284009... 


23 k
e 2 2 cos 3

 
3
3e
k  0 (3k )!

32
1
16 log 2 
 k
3
k  0 2 ( k  4)
J803
1
.423691008343306587163... 
1
  log 2  CosIntegral 2    log x sin 2 x dx
2
0

.423793303250788679449... 
1 .4240406467692369659...
csc 7 
5

21
(1) k 1

2
k 3 k  7

2 7

1


   1 

( k  1)( k  4 ) 
k 1 
.424114777686246519671... 
24   2  6 (4)  6 (3) 

k
k 2
5
4
 k4
1
=
 log(1  x) log
3
x dx
0

.424235324155305999319... 
  (k )  1
k
k 2

1 .424248316072850947362... 
 (1)k 1
 (k  1)
k2
 1 
4

 
3
  1/ 2 
k 1
.424413181578387562050... 
.4244363835020222959...

=

 1
1/ 4 erfi   
 2
2e
1
Dawson  
 2

1 1
   Li2  
k
k 1 k
( 1) k 1


 2k 
k 1
k ! 
k


e
0
 x2
sin x dx
GR 4.381.1
.424563592286456286428... 

21
tan
  (k )

.424632181169048350655... 
 21
2
( k ) 1


7

15
k
k 1
2
1
 5k  1

1
k 2

14 .424682837915131424797... 
.42468496455270619583... 
x 2 dx
12 (3)   x / 2
e 1
0

  log 2  1
2
log x sin 2 x
dx
x2
0
 
(1) k 1 (4k  2)


2k  1
k 1

1 .424741778429980889761... 


1
 arctan k
k 1
2


=
i
 1 i 
 1 i 
 1 i 
 1  i 
 log 1 
  log 1 

  log 1 
  log 1 
2
2
2
2
2  




.424755371747951580157... 

32


 
=
 ( 1)
k 1

9 .424777960769379715388... 
csc h 3 8 2 cosh   cosh 3  12 sinh   cosh   
3
k 1
k 2 ( k 2  1)
k  (2 k )  1   2
3 
k  2 ( k  1)

2
.425140595885442408977... 
=
.425168331587636328439... 
1 i 3 
1  i 3 
2 
 

2  2   

 2 
3 
2






k 1
(1) k 1  (3k  1)   (3k )    3

k 1
k 1 k  1


e
2


 (1)
k 1
k 1
3 .425377149919295511218... 
.425388333283487769500... 
2
dx


.4252949531584225265...
cos 2 x
 1 x
 coth

H 2k
1
3
1 
  log  2 arctan

k
2
3
2
2
2
2 log 2  2 log 2 
2

Hk
3
k
 4k
k 1
.4254590641196607726...
.4255110379935434188...


2 .42562246107371717509... 
csch1
e
 2

2
e 1

Bk  2 k 
 

k 0 k !  k 
e
k 0
1
2 k 1


22 7
3k 2  3k  1
  2 ( 3)  2 
3
3
4
k  2 k ( k  1)

=

 ( 1)
k
k ( k  1)( ( k )  1)
k 2

.425661389276834960423... 
H ( 2) k

k
k 1 3 k


log 3
1
 

4
12 3
k  0 (3k  1)(3k  3)


1
k  ( 2 k  1)
.42606171969698205985...   ( 1)


3
k
2
k 1
k 1 2 k  k
1 
i 
i 

=     1 
   1 


2 
2
2 
.425803019186545577065... 
.42612263885053369442... 

4
2 
e
(1) k


k
k  0 ( k  2)!2


8 .4261497731763586306...

71
root of  (1) ( x)  1

1
2 .42632075116724118774...   2 
k  1 Fk
1 .426255120215078990369... 

GR1.232.3, J942

.42639620163822527630... 
log k
 k (2k  1)
k 1
 5  1
3 5
  Li2 

 log 2 

 2 
15
2




1

(1)
.426473583054083222216...  
k  0 7k  2

8 2 log 2
1
.426790768515592015944... 

 
9
3
k 1 k ( 2k  3)
.426408806162096182092... 
1 .426819442923313294753... 
.426847730696115136772... 
2 .427159054034822045078... 
.427199846700991416649... 
2


 
2
8
 log 2 

J265
1  dx
2
 x
 log(1  x) log1  x
0
4k  1

2 
k  2 2 k ( 2 k  1)

3

2
k ( ( k )  1)
2k
k 2


2 2 log 2  1
2 cos 2  2 sin 2

4
2 1 


12
8
.42721687856314874221... 
.4275050031143272318...

3 (3)
G 

8
Berndt Ch. 9
 /4

0

 (1)k 1
k 1
2k k 2
(2k )!
4
cos x
dx
1  sin x

5
 2
 2 log 2         
3
6
 3
k 1

(1)
= 

2
k 1 3k  k

 1  x2 
dx 
=   log
2 
1 x  x 
0


=

1
 log1  ( x  1)
0
1 .427532966575886781763... 
3

 dx

9 2
1


Li2  e3i  Li2  e  3i
4 12
2
 


   (1)

k 1
k
cos 3k
k2
GR 1.443.4
2 2 1
  
2
 2
.42772793269397822132... 


k 0
(1)
2k  2
k
1
.42808830136517602165... 
 x tan x dx
0
.428144741733412454868... 
 (2) log 2 
log3 2  (3)


3
2

H ( 2) k


k
k 1 2 ( k  1)
2 .428189792098870328736... 
1
cosh

 3
2

 (1)
1
1/ 3
(1)
2/3


1
3

  1  k

k 1
Berndt 2.12

=

 1  k
k 1


403 .428793492735122608387... 
.42881411674498516061... 
1 .42893044794135898678... 

 cos
e6 
 2 1 log 2 log 2 2
 


12 2
2
2
   3 
     1 
2   2
2e  17

54
3
.42909396011806176818... 
.429113234829972113862... 
1 .42918303074795505331... 
.42920367320510338077... 
 
=


( 1) k


k  2 k !( k  1)

log(1  x )
x ( x  1) 2
0
1


x 1/ 2
0 e x  e x  1 dx 

3k

k  0 ( k  3)!
2
23
 (1  2i )   (1  2i )
2


1 

k
e
  Re i e
2

1
k
.42828486873452133902...   log
k 1
k  2 k!

(e  1)(log(e  1)  1)
Hk
.428504222062849638527... 
 
k
e
k 1 (e  1)
3
.428571428571428571
=
7

( 1) k 1
1
 .4287201581256108127...   
 Ei ( 1)  

e
k 1 ( k  1)! k

( 1)k
 
= 

k  2 ( k  1)!  2 k !
.42821977341382775376... 

2

2

1


2
1
k 1 4 k  4
( 1) k


3
k 0 k  2

( 1) k 1 ( k  12 )!


( k  12 )!
k 1
k

k 1 2
= log 2   ( 1)
( k  1) 
2k
k 1


 

 
=
GR 8.373


 
=

2k
  2k 
k 1

 
(2 k )!!

k
k (2 k  1)
 (2k  1)!!2

k 1
  (2 k  1) k
k

sin 4 k
=  

k
k 1
1

 
=
 arccos x arcsin x dx
0
 /2
=

0
x  sin x
dx
1  cos x

=
Prud. 2.5.16.15
dx
 (1  x ) cosh x
GR 3.522
2
0
.42926856072611099995... 
6 .429321984298354638879... 

 
=
4 .429468097185688596497... 
.429514620607979544321... 
.429622300326056192750... 

 

 



 2   1 
log  2    1    
3
 3   3 


e
x
0
log x
 e x  1
GR 4.332.2
 



 
 2 cot
  2 2 cot
csc2
 3 csc2

16 
2
2
2
2


2 k 2 (2 k 2  1)
2  (2 k )
k



2
3
2k
k 1
k  1 ( 2 k  1)
3
7
35

256

 (x
0
dx
 1)5
2
1
9
 1
 3
Li3     Li3     (3)   3 log 2 log 2 3  3 log 2 2 log  
6
2
 2
 2
729  3  
1
9
 2
   3i  log 3  log 2 log   log
Li2    
6
2
64

 2 
(1) k 1 H k

2k k 2
k 1

2k  (k )  1
.429623584800571687149...  

k3
k 2

=


 2
2
  Li  k   k  
3

H
1 .429706218737208313187...   I 0 ( 2)  K 0 ( 2)   k
k 1 k!k!
k 2


3 .42981513013245864263... 
4 .42995056995828085132... 

G
2
 3
2
cot  3 


 3  (2k )  1   k
k
k 1
k 2
2
3
3
1 .429956044565484290982... 
 ( 2k )
4
k 1


9

.43020360185202489933... 

2 3
k
k
2
log 2 x
0 (1  x)4 dx


 1 

2 

 Li  4k
2
k 1
( 1) k 1  ( 3k )
.43026258785476468625...  

3
2k
k 1 2 k  1
k 1
2
1
2
1 5
.43040894096400403889... 
arcsinh 
log

2
2
5
5


1
( 1) k 1


k 1  2 k 
 k
k


=
=
(2k )!!
(2k  1)!!22 k 1
k 0


Fk
Fk Lk



k
k
k 1 2 k
k 1 4 k
 (1)
k



 
 .43055555555555555555
=
=
.430676558073393050670... 
dx
1 x3  x 2  x 

e
0
x
J142
dx

 e x  3

31
k2
 
72
k 1 k ( k  3)( k  4)
log 5 2

11 .430937592879539094163... 
48  64 log 2 
x
 1/ 2
0
1 .430969081105255501045... 
 1
Li2 (  x ) Li2    dx
 x
1/ 5
6
(1) k 1

k
k
k 1 2 ( 2  1)

.431166447015110875858... 
3 3 log 2
x yz

 
dx dy dz 
2 2 2
4
2
1

x
y
z
0 0 0
24
4

2
( 2)
2

1 


 coth    ( 1) k  ( k )   (2 k ) 
2
6
2
k 2
3

k 1

2
2
k  2 k ( k  1)( k  1)
1 1 1
1 .43142306317923008044... 
2 .43170840741610651465... 
.431739980620354737662... 
=
3G 

5 .43175383721778681406... 
Hk
F
k 1

5 .43181977215196310741... 
k
x dx
 ( x)
0
.43182380450040305811... 
( 1) k 1 k


2 k 1
( 2 k  1)
k 1 2

1 1
1
 arctan 
5 2
2

.4323323583816936541...

e2  1

2e 2
(1) k 2k


k  0 ( k  1)!

e
dx
1 x3 
1
dx
e

2x
0
1/ 4
  1 1 / 4 
i    1  
 1 i 
 1  i 
 H      H   3 / 4   H  3 / 4  
H   
 2  
4 2    2  
 2 
 2  




1
( 1) k 1  ( 4 k  1)
= 


3
1
2k
k 1 2 k  k
k 1
.43233438520367637377... 

 

.432512793490147897378... 
 ( 1)
k 1
log  (2 k ) 
k 1
205
 H ( 2) 4
144
e
.432627989716132543609... 
2

1
sin x
.432824264906606203658... 
  x
2
1  log 
0 
1 .432611111111111111111 =
.432884741619829312145... 
.432911748676149645455... 
33 .4331607081195456292...
arctan e 
3

4

4
 e

 x4


 arctan
4
 e x
0
1

e
1
e
x
0
dx
 e x
 dxx
GR 3.469.2

1  7 (2)  6 (4)  12 (3) 
=
k (k  4k  1)
(k  1) 4
k 2

 k  (k )  1
3
k 2

1 .43330457796356855869... 
.43333333333333333333
=
2

81 (3) 189


2
4
13

30
1
log 2 x
0 1  x1/ 3 dx

arctan x1/ 3
1 x 3 dx

k2
 (k )  1 
4 .43346463275973539753...  
k 2 k  1
.433629385640827046149...  13  4 
.433763428189524336184... 

3


1  3k  2
 k  (k  1)
k 2

  log 1  9 x 1 1  dx



2 1 
2
0
2
 log
k 

k  1 

e2  1
.433780830483027187027...  log
 log cosh 1 
2e
i 
i

1  1  
  
log 
 
=
 2i   2i 
1  1  
   

2 2 k 1 (2 2 k  1) B2 k
= 
(2 k )! k
k 1
1
=
 tanh x dx
0
.433908588754952738714... 
2 sin  2




 
=
k 1

1

1 .43403667553380108311... 

2

2

 


 1  k (k  2)  

89

k 1
9 .4339811320566038113...

 1  (k  2)



k 1

1

 1 k (k  7) 


.434171821180757177936... 
  (k  1)  1 (2k )  1
k 1
3 .43418965754820052243... 
3 log 
45 79e1 / 4 
1


erf
64
128
2
1
.434294481903251827651...  log10 e 
log10

k !k 3

k 1 ( 2 k )!
1 .434241037204324991831... 



1
 (1) k 1
.43435113700621794951... 
csc
   2
2 3
3 2
k 1 3k  1

1
1 .43436420505761030482...  
k
k 1  2 k

 ( 2 k  1)
.434591199224467793424...  
2k  1
k 1

 2k  1
2e 2
5e 2
1 .434901660887806674581... 
I 0 (2) 
I1 (2)    
3
6
k  0  k  ( k  2)!
 
1 .4350443315577618438...

2 sinh1/ 2 

2
k 0
1/ 2 2 k 1 ( 2 k 1)!


 .4353670915862575299... 
1

 (k  2)!!  k !!
k 1

sin 2 cos 2
( 1) k 4 k k
.4353977749799916173... 

 
4
2
k 1 ( 2 k  1)!
k 5

23
(1) k
.435943911668455273215...

 
k!2 k
32 e
k 1
2  arc coth 2  arctan h 2  3  


.435953490803707281686...
 /6

0
dx

sin x  cos x
1 2 3

, Lebesgue constant
3


2
(1) k 1
.43617267230422259940...  1 
 2 log 2   3
2
12
k 1 k  k
1 .435991124176917432356...
.436174786372556363834... 
1
log(1  e)  log 2 

2e
1
cosh x
 1 e
x
dx
0

1  (1)  2 
(1) k 1
(1)  1  

.43634057925226462320...  9          
2
4
 3
 6   k 1 (k  1 / 3)
.43646265045727820454... 
2
25
csc2
2

5


1
  (5k  2)
k 1
2

1

2
(5k  3) 

k2
.436563656918090470721... 2e  5  
k 1 ( k  2)!

k2
5 .436563656918090470721... 2e  

k 1 k!
( 2)

k
k 1 H
.43657478260848849014...   ( 1)
k 2
2 k
k 1
k 1
k  0 k!



 
 ( 2k )
i  
i 



 2 log  1 
   (1) k 1 k 1 
 1 
2 k
2 
2 
k 1
 

1 

= 2 log 1 

 2k 2 
k 1
log 2 3
2
1 
 log 2 log 3 
 Li2   
1 .43674636688368094636...   Li2 ( 2) 
6
2
3
1 .436612586189245788936... 
1

 
=
log 2 x
0 (2 x  1) 2 dx 
1

 
=

0
.436827337720053882161... 

7

 log1  2e  dx 
x
0
log x
dx
x  1/ 2
tanh
 7
2

3

4

k
k 2
2
1
k2
.43722380591824767683... 
42   2  18 (3)

24
2 .437389166862911755043... 
 (4)   (2)  3 
=
2k  k  k  1
k 4 (k  1)
k 2
1

1
 x log1  x  log
2
x dx
0

 k  (k )   (k  3) 
k 2

.437528726844938330438... 
.43754239163900634264... 

3
2
 csc 14

2 14
257

1820


 (k  1)  12
k 1
k!

(1) k

2
k  4 k  14

(1) k 1 e k H k
.437795379594438621520...  e  Ei (e)    1  
k!
k 1
3 
 5 
 5
 3
16 2  128G  16  2,     4,     4,   
.43787374538331792632... 
128 
 4 
 4
 4

e


 
x3
= 
dx
3
0 cosh x
1
.438747307343219426776... 
.43882457311747565491... 
log x
dx
ex  1
0



log 2
(1) k



4
2
k  0 ( 2 k  1)( 2 k  2)
=
 /4
=

0
.43892130408682951019... 
.439028195096889694210... 
.439113833857861850508... 
 
=
k 1
2
1
 6k  1
x dx

cos 2 x
 coth 
2
k
Prud. 5.1.2.4
 arctan x dx 
0

1

2

 (k  1)  1
k 1
k

J72
1

k
k 1
1
2
 1
2
4  2 (2)  2 coth

2


 4k
k 1
4
2

 k2
 
   (k  3) 
i  i 
 Im4 1   
   Im

k
 k 1 (2i ) 
  2 3 
2

 8k

 (1)
(1) 

1   

2k  1 
k 1  2k

(1) k / 2 

k
k 1
k
=

7
.439255388906447904336...  4 log 2 

3

2


k  2 ( k  1)( 2k  1)
2k 1  1
 (k )  1 
(1)

2k  2
k 2

k
2 ( k )
2 .439341990042075362979...  
,  (k )  number of prime factors of k
k 1 k!



5 .43937804598647351638... 
7 4  2 log 2 2 log 4 2
log 3 x dx
 1


 3 Li4      2
 2 1 x  2x
120
4
8

.43943789599870974877... 

(1) k 1
k2
2
e
5


2 2e
k 1
.43944231130091681369... 
.43947885374310212397... 
1 .43961949584759068834... 
 
k
3 
(1) k 1 k 

2 k 1
2

 1  dx
 cosh x  x
4
1
4 ( 7 )   ( 6)   ( 2 ) (5)   ( 3) ( 4 ) 
 1/

.44001215719551248665... 


 ( k  1)
k 1
k6

1
 k (k  1) log k
k 2
.440018745070821693886... 
.44005058574493351596... 

5 3

J1 (1) 
1 2 log 2


5
5
(1) k

k  0 ( k  2)(3k  1)

1 
(1) k


2 k  0 k!(k  1)!4k
1
1
1
 ( log2  2 cos1  2) sin  (  1) cos  
2
2
2



  (2k  2) 
1 .44065951997751459266... 
tanh
  Im

2
2
ik
 k 1


1
 
= 
2
k 1 2k  2k  1
.44014142091505317044... 


=
sin x
 sinh x dx

1
 k  1 sin
k 1
2k  1
2
GR 3.981.1
0

=
e

2 .44068364104948817218... 
sin x
dx
 e x
x
1
Ei(e)    1 
e
.440865855581020082512... 
1  J0 ( 2 ) 
.440993278190278937096... 
 3
4 2  2 
3
 

ek

2
k  0 k!( k  1)
(1) k 1

2 k
k 1 ( k!) 2

Berndt 2.5.15
3 .441070518095074928477... 
6 (4)  6 (3)  12 (2)  9 

 k  (k  3)  1 
3
k 1
k  4k  1
2
(k  1) 4
k 2

3 .4412853869452228944...
2
=
k

3
 3 1
 3
      25 / 4  3 / 4  1      sin
8
 4
 4  4

.44133249687392728479... 
k
k 2
5 .44139809270265355178... 
2
1
(k  1) log k
 3 

 (k 
k 0

1

1 )( k  2 )
3
3

=
log( x 2  3)
0 x 2 dx 

3 .441523869125335258...

1 .441524242056506473167... 

e I 0 (1) 
1  2k 

k 
k
 k !2
k 0
3 (3)  2 (4)
5 5
5
5

arccsch 2  log 2 
log(161  72 5 ) 
10
4
16

(1) k
 
= 
k 0 k  1/ 5


1
1

k
.44195697563153630635...   ( 1) Hk ( ( k  1)  1)  
log 1  

k
k 1
k 2 k  1
1/ 3
1 .44224957030740838232...  3
4 .44156786325894319020... 

5 .44266700406635202647... 


( 1) k 1
.442668574102213158178...  

k  1 k ! k ( k  1)

1
1 
log 2
.44269504088896340736... 
 

1 .44269504088896340736... 

2
k 1

1  2   3 1  2
1/ 2 k
k 1
k

1/ 3k
k
 22 / 3
k


[Ramanujan] Berndt Ch. 31
1
log 2

.44287716368863215107... 
k
2  21/ 3

1
k

3
k 1 ( k  1)
 (2)   (3)  

 (1)  (k )   (k  2) 
k
k 2

=
Hk
 k (k  1)
k 1
.44288293815836624702... 
4 .44288293815836624702... 
2
1
arcsin x
dx 
1 x
0

 24 
 1   3
 4  4
 2      



=
ex / 4
 e x  1 dx



 
=
 log1  x  dx
4

 log1  2 x  dx
2

0
0

=
log( x 2  2)
0 x 2 dx

=
dx
1  sin
2

x
 /2
=

tan x dx
0
.44302272411692258363... 
.44311346272637900682... 
log tan 1 

4
 /2
=


(1) k 1 22 k (22 k 1  1) B2 k

(2k )!k
k 1

x e
2  x2
0
xe  tan
0
2
x

dx   xe  x dx
4
1  cos 2 x
dx
cos 4 x cot x
1

4
(1) k k

2
k 2 k  1
.443189592299379917447... 

3
cot  2 

4 2 2
 (1)
k 2
1 .443207778482785721465... 
k

k
k 1
.443409441985036954329... 
2
1
 4k  2
Lk  2  (k )  1
2 (3)  2 log 2 2 

(k  1) H k
2
(2k  1)
k
k 1
.443259803921568627450 =
GR 3.964.2

log 2 
=
LY 6.30
0
.443147180559945309417... 

AS 4.3.73
3617
  (15)
8160
1
1


1 / 2
1/ 2
2 cosh 1 / 2
e e

 (1)
k 0
k
e  ( 2 k 1) / 2
J943

1 .443416518250835438647... 
 (k )  (k )  1
k 2


3
2 HypPFQ 1,1,1, , 2,2,2,2,4  
2


1 .443635475178810342493...  arcsinh 2

1
( 1)k 1
.4438420791177483629...    log 2  Ei(  )  
k
2
k 1 k ! 2 k
3 .443556031041619830543... 
.444288293815836624702... 
.444444444444444444444 =

5 2
4

9

=
5 .4448744564853177341...

k 1
e1 / e  1  


 2
log 3 xdx
 
 2 (3)   
ex
2
0
3
 ( 1) k k ( k  1)
( k )



8k 2


3
k 1 ( 2 k  1)
3
9


3 log 3
1
.44518188488072653761...  3 


  2
2
2 3
k 1 3k  k

 4
 1
k  1  ( k  1)
 
=       hg     ( 1)
 3
 3
3k
k 1
1
=
  log(1  x 3 ) dx
0

k 2H
48
42
6

log   k k
125 125
5
k 1 6

3
3
(k  1) H k
.445901168918876713517... 
1  log   
4
2
3k
k 1
.445260043082768754407... 
2
1  log x
dx
xx
2k
 (k )
2
1 .44494079843363423391...   ( 3)   0 3
k
k 1

( 2 k )
1 .44501602875629472763...  
k 1  ( 2 k  1)


e1 / e , maximum value of x1 / x

1

k
k  0 k!e
k 2
3 .44514185336664668616... 

k

1 .44490825889549869114... 
k 1
k
4
0
1 .44466786100976613366... 
 2k  1
  k  k!k

1/ e
.44466786100976613366... 

Titchmarsh 1.2.1
1 2
1

 2 (2)   (4)   Li4 (e 2i )  Li4 (e  2i ) 
3
3
2

cos 2k
= 
GR 1.443.6
k4
k 1

(1) k
.44648855096786546141...  2(cos1  2 sin 1  2)  

k  0 ( 2k  1)!( k  2)

.446483130925452522454... 
 (k )

.446577031296941135508... 
 (k !)
k 2


.446817484393661569797... 
 1 
2

  2 
1
  1 
0
k
k
k 1


  I 
 (k  1) 

 (k ) 

k 1
 
k  2 dk
k 2

2 .4468859331875220877...
.44714430796140880686... 
=
2  cos 2  2 sin 2
Li3 (ei )  Li3 (e  i )  
(1) k 1 2k


k 1 ( 2k )!( k  1)

5
5
128 32G 4  2 24  144 log 2
.447309717383151082397... 





27
3
3
27
27
1
log(1  x ) log x
 
= 
dx
x 1/ 4
0
.447213595499957939282... 

.44745157639460216262... 

3 3

 2  21/ 3  

 (x
0

2
3
dx
 1)( x 3  2)
3
1
.447467033424113218236... 
   (1) k k  ( k )  1 
12 8
2 k 2
 2
2 / 3  1  1  2 
2 .4475807362336582311...        ( 2 )      
 3  3
 3
 3
.44764479114467781038...   tan
2


( 1) k
1

.44784366244327440446...   k
  log 1  k 

3 
k  1 ( 3  1) k
k 1
.447978320832715134501... 

12


2
( 1) k 1

2
4
k 1 k  k

=
coth

k
k 1
3
1

 2k 2
i

  i 3 

csc (1)5 / 6  csch
 tanh 
6
6
2
12
2
3 (3)
3 .448019421587617864572...   (2) 

2

H k H k 1
 k (k  2)
k 1
1
e e
 e  coshcosh 2  sinh 2   sinh cosh 2  sinh 2
2
2
k

e cosh k
= 
k!
k 0

  
cos( 2 k  1)
.448302386738721517739... 
GR 1.444.6
  1  
2 
42 
k  0 ( 2 k  1)
log 2 2
log 2 3
 1
 1
.44841420692364620244... 
 log 2 log 3 
 Li2     Li2    
 3
 2
2
2
e2
810 .44813687055627345203... 
=

 
=

 
=

 
=
1 2
 2 
   3 log 2 2  3 log 2 3  6 Li2   
6
 3 
2
log 2 2
1 1

 Li2   
12
2
2  4
k

(1)

k
k  0 2 ( k  1)( 2 k  2 )
1
log 2 x dx
0 ( x  2)2 

(1) k 1


k 2
k 1 2 k


Hk
3 k
k 1
k
1
log 2 x dx
log x dx
1 (2 x  1)2   0 x  2


 
=

e x 
log

1

0  2  dx 

k (2k )
2  2 


 2  5 sin
 csc
40 
5k
5
5
k 1


 1
k 1  ( 2k )
Li2   2 
1 .448526461630241240991...   ( 1)


2
k
 k 
k 1
k 1
k


1
(1)
 
1 .448545678146691018628...  
k 1 k! H k
k  2 S1[ k , 2]

1
cos k

Li3 ei   Li3 e  i    3 
.44857300728001739775... 
2
k 1 k
.44851669227070827912... 
 




1 

.448623089086114197229...   1 
  2k  
k 1
  k 
  k  

9
(1) k
.448781795164103253830... 
6  
k
e1 / 3
k  0 ( k  2)!3


5k 2


2
2
k 1 (5k  1)


5/ 2
x dx
.44879895051282760549... 
 
7
1  x7
0

32 
( k  12 )!( k  12 )!
6 .44906440616610791322... 
 

( 2 k  2 )!
9 3
k 1
 2
.449282974471281664465...  Li2  
 5
.449339408178831114278... 
1
1


2 2 2

dx
x
3

1

2
log 2
.449463586938675772614... 



9
18 2 9 3
2 .449489742783178098197... 
110 .449554966820854649772... 
.44959020335410418354... 
3 .4499650135236733653...





 (x
0
3
dx
 1)( x 2  2)
6
41e  1 
1

2 2

k6

k  1 ( k  1)!
cot

2 2
9 2  6  2 


 4k
k 1
2
1

 1/ 2
3
  2  4 (3) 
2
2

x 2 log 3 xdx
0 e2 x 
.45000000000000000000
=
3 .45001913833585334783... 
.45014414312062407805... 
9
20

 2  log 2 2
log 2 (1  x) x log x  x  2
 
dx
x
( x  2) 2
0
3
log 2 
 3
3
k

1 2
2 .45038314198839886734...   ( 1)
k!!
k 1
.450571851280126820771... 
1 

2
2


cosh
2


sin
( 1) k


k  0 ( 2 k  1)( 3k  2 )


 sinh

cos

2
2
2
cos 2  cosh  2
(1) k 1

4
k 1 k  1
3  cos 2 sin 2


.45066109396293091334... 
2 

GR 4.313.7
2


=
( 1) k


1
k  0 ( k  2 )!( k  2 )

2
 2k 
1
 


k
k  0  k  16 ( 2k  3)
. 45071584712271411591... 
2G  1

2
.45077133868484785702... 
3 (3)  (1) k 1


3
8
k 1 2k


=
x 3 dx
0 e2 x  e 2 x  2
1

 
=
 2 Li3 (i )  Li3 ( i )    Li2 (  x 2 )
0
4 .450875896181964986251... 

6
216

.45091498545516623899... 
.45100662426809780655... 
.45137264647546680565... 


  3 (2) 
(2 k )
k 1
3
8
dx
x
2k k
1
 3  6 
 arcsin
3
x dx
0
1  2
  
3  3

x dx
e
x3
0

3 .451392295223202661434... 
.45158270528945486473... 
 log 3 
log

2

log( x 2  4)
0 x 2  1 dx

 (2 k )
k 1
4k k


1 

   log 1  2  
 4k 
k 1
Wilton
=
=
Relog log i

2k  1 
 1
 log



2k 
k    2k
Messenger Math. 52 (1922-1923) 90-93
Prud. 5.5.1.16
k 0
1  x dx
1  x log x
1
=

0
 /2
=
1
GR 4.267.1

  x  cot x  dx
0
GR 3.788



1  i
1 i
1  i
i e1  i
sin k sinh k
 
e  ee  ee  ee
4
k!
k 0
k

1
(1)
  3 log 2  3  
.451885886874386023421... 
12
k  0 12k  2

2
2k k 2
 (k  1)  1 
2 .451991067287107389384... 
 1  log 2  
3
k 1 k  2
1 .451859777032415201064... 



 
.4520569031595942854...


.45224742004106549851... 



k 2


k
  k log k  2 
 (3) 

2 (k (k  3)  1) 

( k  2) 2

3
4
p 2 
p prime

(k )
k 1
k

log ( 2 k ) 
Titchmarsh 1.6.1
(2k  1)  (k )
2
2
k  2 k ( k  1)

=

Shamos

3  3
2
(1) k 1

csch 
  2
4 2 2
3
k 1 k  2 / 3

1
.452404030972285668128... 
 coth   2   (i )   (i )    (4 k  2)   (4 k  1) 
4
k 1

1
 
=  2
1
k 2 k  k  1  k
.45231049760682491399... 

16 .452627765507230224736... 
.452628365343598915383... 

2 2 k 1

2
k  0 k !( 2 k  1)
1   1 1
 1 5 
  ,     ,   
 2 6 
6   2 3

.452726765998749507286... 

  (2k )





k 0
(1) k
3k  2
   (3k )  1
k 1
.45281681270310393337... 
erfi 2 
1
2 2
5
5
5 log 2
5
log



16 8 2
2
2 2
8 2
(1) k 5k  (k )


8k
k 2

=
.452832425263941397660... 
.4529232530212652211...

log
2e

3
e
25
k 1
x
x
0
e
dx
2
1
1

2i
2  e  / 2

1 .452943571383509653781... 
1

 8k (8k  5)
  3
( k ) 1
k 2
 1 

.453018350450290206718... 
.45304697140984087256... 
9 .4530872048294188123...

.453114239502760197533... 
.453344123097900874903... 
.453449841058554462649... 
.453586433219203359539... 
2
sin x / 2
 
2
ex
 4
0
e
6

16 
sin 2 ( 4 x 2 )
 
3
x4
0
1

1  log 2 3

4

sin x
x
0 3



si ( x) cos 2 x
dx
0
x


dx
1 4 x 2  3 
(1) k 1


4
k 1 k  k

 8k
pounds/kilogram
.453602813292325043435... 
 cot
.45383998041715262...

x dx
4
3
x
0
k 1
2
k
2
2 2
7
 8 log 2  2 2 log

8
2 2


1
 (k )
  3k  4

2
k 1 4 k  k / 2
k 2 2
 /4
.45383715497509933263... 

4 log 2 1  
 (3  i 3)
 (3  i 3 

   cot
 cot
3
3 6
4
4

.453592370000000000000 =
=

dx
  2

3x  4
4 3
0

=
GR 3.852.3

sin x dx
0


n 2
k 1
 (1)n  log (nk )
.4538686550208064702...

.45395280311216306689... 
 i   i
cot(log  )  i
i  i i

3    5 
     1 
4   2 
x 3/ 2
0 ex  ex  1 dx
5 2 log 2 2
1  1 i 
1  i 

  Li2 
  Li2 
  
96
8
2  2 
 2 

1
 1
.454219904863173579921...  Ei     log 2  
k
 2
k 1 k ! 2 k
.453985269150295583314... 
4 .45427204221056257189... 
2 cosh  sinh  csch 2 
k2  4


2
k 1 k  2


 
=

.454313133585223101836... 
 (1)  2
k
k 2

.454407859842733526516... 


.45454545454545454545
1 .4546320952042070463...
.4546487134128408477...
=


=

 1  k
k 1
( k ) 1
2
2 

 2
 1

 (k )  1
k 2
2 .45443350620295586062... 
( 3)  1

(4)  1
5

11

F3k  2
k
k 1 6

2 ( 3)   ( 2 ) 



 ( k  1)
k 1
2k
k2

2
4 

  1  2 2  
(2 k  1)! k 1   k 
k 0
 0 

sin 1 cos1  
 2 / 
sin 2

2
 ( 1) k

.454822555520437524662... 
k
6  8 log 2   k

k 1 2 (k  2)

3 .454933798924981417101... 
 (2k )


1
 B (k )
k 1
2
1/ k 2
e
2
k 1 k
 (k  1)!  
k 1
.455119613313418696807... 
1
log 9

1 .45520607897219862104... 
.45535620037573410418... 
AS 5.1.10
 2  7 (3) 
log2 x
1 ( x  1) 2 x dx
i
 3 

 3 
  ( 2  i ) 
    i  (2  i )   



2i
4  sinh    2 
GR 1.431


 
=

e  x sin x
ex  1
0
dx 

 e
3
.455454848083584498405... 

 ( x ) 1
 1 dx
2

.455698463397600878...


( k  1)
k
4 k
k 1


1
4k
 k log 4 k  1 

k 1
i i / 2  e  / 4 
2
.4559418900357336741...  1 
sinh 
9
1
.455945326390519971498... 
2   2 q squarefree with an odd number of factors
2
q
.45593812776599623677... 
Berndt 5.30
.456037132019405391328... 

1
1 i
1 i
1 i
  i tan
  4 csc
 
 i tanh
 2 
 4 
 4 
8
1
1 i
1 i
 1 i
  4 csc h
  i cot 
  coth
 
 2 
 4 
 4 
8
 


 
=
979 .456127973477901989070... 
.45636209905108902378... 
.45642677540284159689... 
1 .456426775402841596895... 
.456666666666666666666 =
.45694256247763966111... 
.457106781186547524401... 
e  x sin x
 1  e 2x dx

sinh  2
2
 2
  1  2 
k 
k 1 

1
 3
 cos

2
2


3
 1  (2k  1)
k 1
2





 ( 2k )
1
1
 2
  (1) k 1 k
3
2 3
3 2 k 1 3k  1 k 1

1


1

coth
  2
2 2 3
3
k  0 3k  1
coth




137
H5
1

  2

300
5
k  1 k  5k


1
 ( 2k )
 k

2k
k 1 3  1
k 1 ( 3 )
1
1
 
2 4
 /4

0
cos3 x
dx
1  sin x
2 .45714277885555220148... 
.45765755436028576375... 
.45774694347424732856... 
2  log 2 
 cot 2 
( k  12 )!
k !k
k 1


 3  3
 3

log 2  1 1   3  3 
3
 
     
 
  
3
12   2 
 2 
 2 
 2  
(1) k
=  3
k  2 k  3k
2
.457970097201163304227...  1 
cosh 
G
.457982797088609507527... 

2


 
=
 /4
 /4
0
0
  log(2 sin x ) dx 

=
e
.45860108943638148692... 
 ex
2
  
=
1

 
12 2 sinh  2
4
48

3
4
2 
 ( 1)k 1
k 1
.458665513681063562977... 
.458675145387081891021... 
2



Adamchik (5), (6)
x dx
x 2
0
.458481560915179799711... 
 log(2 cos x) dx
sin 4 k
k4
7 ( 3)
  (2) 
4
1  log( e  1) 
(1) k 1

4
2
k 1 k  k

4 1
 
3
2
k 3
k ( k  1) ( k )

2k



 (1)
k 1
1


k
k 1 e k
e
0
dx
1
J157
x


 
=
2 .458795032289282779606... 
Bk

k 1 k!k

6 ( 4 )  12 ( 3)  7 ( 2 ) 
2 .45879503228928277961... 

9

8
 (1) k  (k )  1 
9

8
 ( 1)
k

3
k 2
2
7 ( 2 )  12 ( 3)  6 ( 4 ) 

k 2
8k 3  5k 2  4 k  1

k ( k  1) 4
k 2

=
.45896737372945223436... 
3
8k  5k  4 k  1

k ( k  1) 4
k 2

=
[Ramanujan] Berndt Ch. 5
cos(1  sin 1) (coshcos1  sinh cos1) 
k
k 3 ( ( k )  1)
=
22 .459157718361045473427... 
2 .459168619823053134509... 
.459349015616034745...

.45936268493278421889... 
.4595386986862860465...


 .4596976941318602826...



=


=
e i 2i  e i
i
e
 ee 
2


  (kcos 1k)!
k 1
e
Not known to be transcendental




1 2 2 
2 


4K 
8 
 2  
 2 
2

K 

 2  

1
 1  dx
 sin x  x
2
1

1 .459974052202470135268... 
1

  (k )


 k   (k  1)  1
k 2
.460047975918868477...
2
( 1) k
  3 
 6 log 2   3
2
12
k  1 3k  k
.46007559225530505748... 
2  2 4

=
 (x
0
 /2
=

0


 
=
.460265777326432934536... 
2



2


2 2
 /2

0
cos 2 x
dx 
1  cos 2 x
1 

log 1  2
 dx
 8x  1 
0
9e


dx
 1)( x 2  2)
sin 2 x
dx 
1  sin 2 x
GR 6.151
0

7 ( 6)
 2 ( 3)
 ( k  1)
  (5) 
 
k5
4
2
k 1

1
1
( 1) k 1
sin
 

k
2
2
k 1 ( 2 k  1)! 2

1 

1  k 2  

2 k 
k 1 

1
( 1) k 1
2 sin 2  1  cos1  

2
k 1 ( 2 k )!
e
1
sin log x
0 sin x dx  1 x dx 

3 .45990253977358391000... 
dx
 E (x ) x
GR 1.412.1
.460336876902524181992... 

 (k )
k 2
kk


 1 
erf i
 
2
2 2
 1
1 .46035450880958681289...     
 2
.460344282619484866603... 
e 1 / 8

G
1 .4603621167531195477...
 log 2
4
=

e
 /2


0

=

0
1

 
=

GR 4.295.14
x
dx 
 2 e x  2
x
0
=
log( x  1)
dx 
2

1
x
0

2

=


log x  1
dx
1  x2
1

(1) k

2 k 1
k  0 ( 2k  1)!!2

cos x  sin x
dx
cos x  sin x
GR 3.796.1
arccot x
dx
1 x
arcsinh x
0
1- x2
GR 4.531.2
dx
 /2
=
 log1  tan x dx
GR 4.227.10
0

1 
1 
dx
1 .46057828082424381571... 
   2 arctan
   2
7
7
0 x  x2

tan k
.460794988828048829116...  
k 1 k !


1
(1) k 1 (k  1)!
3  1
.46096867284703433665... 
HypPFQ 1,1,  , 2,    
2
(2k )!
2  4
k 1

.461020092468987023150... 


 i 31/ 6  32 / 3  
 i 31/ 6  32 / 3   
1 
1/ 3
2/3




  (1)     1 
  (1)     1 
   
35/ 3 
2
2







1
 ( 3k  1)
(1) k 1


2
1
3k
k
k 1
k 1

tanh k
1 .46103684685569942365...  

k!
k 1
1
1  (1) k (2k )!
.46119836233472404263... 
sin  
sin( 2k  1)
2 k  0 (k!) 2 4k
2 cos1

=
 3k
Berndt 9.4.19

1 

.461390683238481623342...   1  4
  lim
j 
k  1
k 1 
( j!) 4
 k
j
4


1
k 1

.461392167549233569697... 
3 
1  dx
 cos x  1
   


2
 x
4 2
2 ( x  1)  x
0
GR 3.783.1

5 x
 x e log x dx
2
=
0
.461455316241865234416... 
1 .4615009763662596165...

e  1
Ei  
2  2

2 ( k )

k 2 ( k )  1

k log  (k )
k 1
k 2

1 .46168127916026790076... 
.46178881838605297087... 
.461939766255643378064... 
.462098120373296872945... 

9  2 3  12 log 2

30
x
0
4
1

log 1  3  dx

x 
2 2
3

 cos sin
4
3
8

2 log 2
(1) k
 
3
k  0 ( 3k  1)( 3k  2 )
1

log1  3 dx
x 

1
=
1
x
2
0
log(1  x 6 )
dx
x4
0
1

 

=
.462117157260009758502... 
36 .462159607207911770991... 

e1
1
( 2 2 k  1) B2 k
 tanh  2
e1
2
( 2 k )!
k 1

4 2
8 ( 2 )
27
9



1
1

=  

2
2 
(3k  2) 
k 1  (3k  1)
1 .462163614976201276864... 
 G2 
AS 4.5.64
J309
1
=
log x dx

x3  1
0

1

 
=
2
1 x
1 x
0
3
log x dx 

1 .46243121900695980136... 
1
p


  1  2
p prime
1 .462432881399160603387... 
.46260848376583338324... 

 (k )
k 2

k !!


   (k )  1
1/ k
k 2
1 .4626517459071816088...

=
2
1
 
2

1
 k !(2 k  1) 
erfi 1 
k 0



1 
1
p(k )

3 .462746619455063611506...   1  k

  

k
k
2 1
k 1 2
k 1 
k 1 (1  2 )

5
( 1) k
.46287102628419511533...  20 log  4   k
4
k  0 4 ( k  1)( k  2)
1
.4630000966227637863... 
1   2 csch 2  

2


 
=
 (1)
k 1


 
=
k 1
k  (2k )   (2k  2)  
k (k  1)
 Re  (1) (1  i )
2
 1) 2

 (k
k 2

=
320 .46306452510147901007... 
.463087110750218534897... 
xcos x
0 e x  1 dx 



xcos x
dx
x
1
 e e

x
0
6
3
  1
3/4
 
4  3
3/ 4 
   1 1 / 4 
 cot       i cot    1   
 31 / 4  
   3 



 


 
=
=
1
 k 2
k 1
k 1  ( 4 k  2)
  1
3k
 3k
2
1
1 5
.46312964115438878499...  2 arcsinh
 2 log 2

2
2
2
3 .463293989409197163639... 
.4634219926631045735...

6 
1
e Ei ( 2)  Ei ( 1)  
e
0
55 .46345832232543673233... 
( 1) k 1

k 1  2 k  2
 k
k
319735800

5764801

k
8
k 1
8
k

x
dx
(1  x )

6
13
5269
1 .463611111111111111111 =
 H ( 3) 5
3600
1
1
.46364710900080611621...  arctan
 arcsin

2
5
i 


 
= Im log( 2  i )  Re i log 1    
 2 

k

( 1)
 
=  2 k 1

( 2 k  1)
k 0 2
.463518463518463518



=

=
( 2 k )!!
 (2 k  1)!! 5
k

k



2
=  arctan
2 
 (2 k  3) 
k 0
k 1

=
 ( 1)
k 1
k 1
.4638805552552772268...

3 .464101615137754587055... 


2
arctan
2   [Ramanujan] Berndt Ch. 2, Eq. 7.5
 (k  1 /  ) 

e
 e
  ( 1) k 1  

e   k 1
k
12  2 3

1
.46416351576125970131...   k

k 1 e  1

(1) k
.464184360794359436536...   3
k 2 k  6
1 .46451508680225823041... 
[Ramanujan] Berndt Ch. 2
(1) k 1


k
k 1 e  1

Berndt 6.14.1
9   (2)   (4)  4 (3) 

 (1) k  (k )   (k  3)
k
2
k 2

k
.46453645613140711824...  9e  24  
k 1 k!( k  4)
24 .46453645613140711824...  9e
1 .46459188756152326302... 


 .4646018366025516904...

.46481394722935684757... 
=
 1/ 3 
5 

4

AMM 101,8 p. 732
 3

1

2
 k 1
3
k 2
 5  i 3 
i 3   5  i 3 
 

 
 2 
3   2 


tanh
4
 
3
k
2

=
  (3k  1)   (3k )
k 1
.464954643258938815496... 


 
=
1
3
 log 
2
2
2 arctan

1
 log1  2( x  1)
2
0
1 .465052383336634877609... 
2
sinh

1/ 3
.46520414169536317620... 
e
 0 
 
 
2
 i / 2

3



.465461510476142649489... 
3 .465735902799726547086... 
k 1
k
2
 
2
(k )   (k )  
 
2
 (k )
(k )  1  
k (k  1)

k 2
.465370005065473114648... 
0
k 2
3 ( 3)  2
2 log 2 

1 
4
12
3


1 log (2  3 )


3

(1) k 1

4
3
k 1 k  k

Discr. Comp. Geom 22:105(1999)
5 log 2 

(1) k  2k 
I 0 (1)
I 0 (1)
 
.4657596075936404365... 


k 
e
e
k  0 k!2  k 
1
(1)
(1)
.466099528283328703461...    1   ( 2  i )   ( 2  i )  i ( 2  i )  i ( 2  i ) 
2


3k 2  1
k 1
 
=  ( 1) ( 2k  1)( ( 2k  1)  1)  
2
2
k 1
k  2 k ( k  1)


1 
2

  (k )  1
k 2

2 .4653001813290246588...

k
k 1
k 2

1 .4653001813290246588...

 1  4k
 k!3

.4653001813290246588...

 dx



(1) k


k
k  0 2 ( 2 k  1)( 2 k  2 )


.46618725267275587265... 
 ( 1)
k

 ( k ) ( k )  1
k
k 2

1
1
  k 
 1
k 1 

1 

 
=  1 
k  
2 
k 0 

1

1 .466942206924259859983... 
  k
  1 k 0 
 2
1
.467058656500326469824...  Li2    Li2   
 3
 3
.466942206924259859983... 

Prud. 6.2.3.1
1    1

k 
k 1 
6 log 2 
24 (2)
 1



 2 
, Porter’s constant
3
log
2
4

2
2
 

 2


=
1 .467078079433975472898... 

1

  log1  k   log 1  k  
1
.46716002464644797643... 
 arcsinh x dx
1  2  arcsinh 1 
0
.467164421397788702939... 


 (2 k )   ( 2 k  1)
k  1 1/ k 2
e 1
k!
k 1
k 2 k

2
 ( k  1)
.467401100272339654709... 
 2   (1) k 1 k

4
2k
k 1

( k  1)( ( k )  1)
 
=


2 k 1
k 2

( k  12 )!
 
= 

1
k 1 ( k  2 )!( 2 k  1)


 /2

 

x
0
2
1 .46740110027233965471... 
2 .46740110027233965471... 
cos xdx   arcsin 2 x dx 

( 2 k )!!
2
( k  1)
 (2 k  1)!! k
k 1
2
  log 2 i  arcsin 2 1 
 
k  (k  1)
= 

2k
k 1
 
k 2 k
(1)


( 2 k )!
k 1



1
2
1 
4
4


0





2


2
k 1 ( 2k  1)

k2


2
2
k 1 ( k  1 / 4)


=
k

=
x dx
 sinh x 
0

=
x sin xdx

2
x
 1  cos
0

=
3 .46740110027233965471... 
Borwein-Devlin, p. 50
log x
0 (x  1)(x  1) dx 
2
1 
4
 1
1 .46746220933942715546...  E  
 4

1
 K (k  ) dk
0
( 2 k )!!
 (2 k  1)!! k ( k  1)
k 1
.467613876007544485395... 

1 .467813645677547003322... 
 2
1/ k 2
k 1

.4678273201195659261...

3 .467891949359644150296... 
1 .467961779578029614731... 
.46804322686252540322... 
3 .468649618760533141431... 
.468750000000000000000 =
.46894666977688414590... 
sin 6 x
0 x5 dx


1 
1 

k2 
 1  2
k 1
1 .467891949359644150296... 

27 log 3
 2 log 2 
16
  2 log 2  1
2

2 
2
8 log 2  2 
coth 3 
7

2
1

18
6
1
I 2 3 
3 1

15
k2
  k 
32
k 1 5


3
 3
   cos

 4
8
(1) k

k 0 k  1 / 4




 (k  3)
2k
k 1

k
k 1
2
1

9
3k

k  0 k !( k  1)!
sin x 4 
0 x 2 dx 

 (2k )   (2k  1)

1
=
32 .46969701133414574548... 
.469981161956636124706... 

4
3
2
21
k
 k ( k  1) log k  1
k 2

log k
3
k
k 1
k 2 k  k




(n)
=    1   H k  ( k  1)  1 
n 1 
k 1


1 
 1
 3
 
=   3
 log k 
2
k  k2 
k 2  k  k

1
2 .469506314521047562476...  
k!
k 1

J124

 Hk 1 (( k )  1) 
k 2

GR 3.552.9



.46921632103286066895... 
2
0
  2 log 2  1

x dx
 (1  x ) sin( x / 4)
 2

2 sinh 
1 

  1 

5
(k  2)2 
k 1 
8
.4705882352941176
=
17

 3 9
(1) k 1
.470699046351326775891... 
  
2
4
k  1 k ( k  2 / 3)
1 .47043116414999108828... 
1 .470801229745552347742... 
 3
coth  3 
2

=
 (1)
k 1

.4709786279302689616...

3
3
3  ( 2 k )  1
k 1 k
1
2
4
21
31

  (3)   (5) 
2
8
2
8
 ( 1)
4

e

 (2 k )
k 1
e
 1 x


k (2 k  1)
k 1
1 .471517764685769286382... 
k 2
2
k

.471392596691172030430... 

k
 3  (k ) 
k 1
.471246045438685502741... 
5

4



k3

5
k 1 ( k  1 / 2)
1
1
2   2  2 k arctan 

k
  log1  k
k 1
dx 
0



2
e
1 e2
sinh 2 k
e  ee   
4
2 k 0 k !
 ( 3)  1
5 .47168118871888519799... 
(5)  1

  (k ) (k  1)

1 .471854360049...
  
 1 
2
k 1   ( k  k )

403 .4715872121057150966...

8 .472010016955485001797... 
1
sinh  csc  2 
2

k 2  2k  2


2
k 1 k  2k  1


20  2 5
4 .47213595499957939282... 
 1
.472597844658896874619...   Li3    
 2
(1) k 1

k 3
k 1 2 k

3   3
 5  3
3 .47273606009117550064...      
  
4  2
 2  2


 
=
e
0
.47279971743743015582... 
dx
x
2/3
1

4 3 1
 
27
3

 (x
0
2
dx
 x  1) 2

x3 / 2
0 e x  e x  2 
1 .47279971743743015582... 
2
4


3 9 3

1 .47282823195618529629... 
 (1)

1
  2k  1
k 1


 (2k )
k 1
( 2 k  1)!
k 1
k



 k sin k 

1
1
k 1
.47290683729585510379... 
( 1) k 1
6  4 cos1  4 sin1  
k  1 ( 2 k  1)! k ( k  1)
.47304153518359150747... 
si (1)

2


1 .47308190605019222027... 
.473095399515560361013... 
.473406103908474284834... 
 1  dx
2
 x
 sin x
1
k 1
( 1)
5
k 1 k  1

.473050738552108261...


3 e
2 
2
k 1

k 
k 1 k ! 2

log 6
  1 
2

k2

k
k  0 ( k  1)! 2
GR 1.212
(1) k 1 2 k
 (k  1)  1

k 1
k 1


3 1
tan 
 
2 5
4 6

 4k
k 1
2
1
 4k  5
  1
7
7 ( 4 )

 
1440
16
2k 4
k 1
9
.473684210526315789
=
19
(4)  1
771 .474249826667225190536...       744 (5)
 2

4
k 1
.47351641474862295879... 
1 .47443420371745989911... 
.47474085555749076227... 
.47479960260022965741... 
7 ( 3)
2
2
 2 log 2 
log 2 

4
4
4

Hk
 (2k  1)
k 1
 2
12
2   (2)   (4)   (3) 

 ( 1)  ( k )   ( k  3) 
k
k 2

=
k 1
5
 k4
k
k 2
3

6
1  2k 
 k  
.47480157230323829219...  2 log
3  3 k 1 6 k  k 
.4749259869231265718...
2
8
 3
 1
  
 hg   hg  
 4
 4
3

sin 2 x
0 (1  sin x) 2 dx 
1
25 / 4  e / 8   2   , Weierstrass constant
 4

1
1 .474990335830026928404...  

k 1 k ! k
.474949379987920650333... 

1
k2
k 1

( k )
.475218419203912909153...  
k
k 1 4
1 .475148949609715735331... 
2
k

35/ 6
35/ 6  
e 3 
34 / 3 / 2

1
3
e

sin

cos

 
35/ 3 
2
2 

1/ 3
.475222538807235387649... 

 ( 1) k 1
k 1
3k k

(3k )!

1 .4752572671352997213...

x 2 dx
0 e x  x 2
 4 (k )

7 .475453086232465307416... 
  (2k  1)
k 2
.475728091610539588799... 
1 .475773161594552069277... 

 1
Ai  
 2
1/ 5
7 

.47591234810986255573... 
 x  (2 x)  1 dx
1
.47597817593456487296... 

 (k ) log  (2k )
k 1
k2

 (3)


k ( k 2  1)
1
  k 2  (2 k  1)  1   2
3
2
8
k 1
k  2 ( k  1)

27 3 (1)  1
k
2
      2
1 .476208712948717496147...   
2 
 3
4
2
k 1 (k  1 / 9 )
k 1
I 0 (2)  I1 (2)
1
  (1)  2k 
 
.476222388197391301...
 1 
 1 1F1  , 2,  4   
e2
2
 k 1 (k  1)! k 
.4760284515797971427...


1 .47625229601045971015... 

 3 (k )
k 2
3
1 .476489365728562865499... 
.47665018998609361711... 
1 .47665018998609361711... 
.4769362762044693977...

.477121254719662437295... 
2k

21

1

1
 
cot  3   2
3 2 3
k 1 k  4 k  1

2

1

cot  3   2
3 2 3
k 2 k  3
 1
erf  1  
 2
log10 3 

5 .47722557505166113457... 

30
3

2
 sinh 
.47764546494388936239...  log



 
=  log 1 
 
2 2
28 .477658649975010867721... 

log 2
.47746482927568600731... 




i  
i 
 1 
 
 
  
(1) k 1 (2k )

 kk
k 1

[Ramanujan] Berndt Ch. 22

k 2 Hk
.477945819212918170827...  8  6 log 2  7 log 2   k
k  1 2 ( k  1)( k  2 )
3 (3)
1   2  i
 2  i
1 i
 1  i 
.478005999517759386434... 
 log 2   
 
 
  
 
 2 
 2 
 2 
4
4  2 
2
( 1) k 1

5
3
k 1 k  k
G
1 G
4 2 

7 (3) 23
(1) k k 2
3 2
 (k )  1
 log 2 

 
4
54
2k
8
k 2


1
k  (k )  1
(1)
  2 1/ k

( k  2 )!
k 2
k 2 k e


1 
 1
 log 2   (1) k 


 k  1 2 k  1
4
k 0



=
.478069959586391157131... 
39 .47841760435743447534... 
.478428963393348247270... 
.47854249283227585569... 
1 .47854534395739361903... 
.47893467779382437553... 
.47923495811102680807... 
.479425538604203000273... 

 
=
.479528082151010702842... 
4 ( 3)   ( 4 ) 

2
csc
1
sin 
2


2 2

 2  Im (1)
1/ 6

(1) k

2 k 1 
k  0 ( 2 k  1)! 2

( 1) k

4 k  0 k !( k  12 )16 k



  ( 1)
J2 2 2 
k 0
k
2 k 1
k !( k  2)!
(1) k 1
.47962348400112945189...  2  2ci (1)  
k 1 ( 2k )!k

 (k )
e1/ k  1
2 .47966052573232990761...  
 
k
k  2 ( k  1)!
k 1

(1) k 1
  2
k 1 2k  1 / 4


AS 4.3.65
.47987116980744146966... 


3 32
1/ 3

x
0

dx

4
3
2
12
F
  kk
25 k 1 4

 (k )  1
.48022960406139837837...  
k  2  0 (k )
.480000000000000000000 =
3 .480230906913262026939... 
1
1

   log log 

x
0
  2 log 2 
dx
1
log
x


Hk
H
2
.480453013918201424667...  log 2   k
 2 ( 1) k k 
k 1
k  1 2 ( k  1)
k 1
GR 4.229.3



 
=
 ( x  1)( x  4 )  dx

(x  2)2  x
 log
0
2 .480453013918201424667... 
2  log 2 
2

1 .48049223762892856680... 
( 2 k  1)
2

2
k 1
GR 4.299.1
k Hk

( k  1)
k
2 k 1
1
( 1) k 1
.48053380079607358092...  
k  1 k ! ( 2 k  1)
k 1

2 .480548302158708391604... 
.48061832131527820986... 
 2  e2 
1
3 (3)   (2)  1 
2
2  ( 2 k )  1

k!
k 1

1 .480645673830658464567... 

k

1 .480716542675186062751... 
Titchmarsh 14.32.3
HkHk
 k (k  1)(k  2)
k 1

 e
2/k 2
k 2
 2k  (3k  1)  1 
k 1


k
k 2

1 
2k

2
3

6 .48074069840786023097... 
42
.48082276126383771416... 
2 (3)

5
.480898346962987802453... 
1

log 8
.481008115373192720896... 
  2arctan 2  2 log 2  log 5  2 arctan
(1) k 1 k!k!


3
k 1 ( 2k )!k


1
5
 2 log 
2
4

(1) k
(1) k 1
=  k

  2 k 1
k (2k  1)
k  0 4 ( 2k  1)( 2k  2)
k 1 2




 
=
0

2 .481061019790762697937... 


 0 (k )
k 1
.481211825059603447498... 

 
=
1
 log1  ( x  2)
k!


2

 dx


1
  ( jk )!
k 1 j 1

1 5

 log  = arccsch 2  log  2 cos  
2
5


cos( k / 5)

k
k 1
log

.48164052105807573135... 
2  2 (2)  4 ( 3) 

1
4 .481689070338064822602... 
e 3/ 2 

k
3
 k !2
k 0
.48180838242836517607... 
.48200328164309555398... 
.482301750646382872131... 
15
6

e
k
1
e
 x1/ 3
dx
0
 (1  log 2)
2
 /2
 
 log sin x tan
2 ( 3)  4 log 2 2 
2 log
4

2
x dx
0

k
k 1
.483128950540980889382... 
log 2 x
dx
x 2 ( x  1) 2


2
Hk
( 2 k  1)
log(1  x 2 )
 cosh x / 2 dx
0

k 2 Hk
15 3
3
2 .483197662162246572967... 
 log   k 
8 2
2
k 1 3
 
1
.483204576426538793772...    (a k )  1 
k  2 a 1 k
7 .4833147735478827712...

56  2 14


log 3 k
    ( 3) ( m)
6 .4834162225162414100...  
k  2 k ( k  1)
m 2
k


(1)  ( 4 k  2 )
1
  sin 2
1 .483522817309552286419...  
( 2 k  1)!
k
k 0
k 1
1
 2 log 2 2 log 3 3 1  1 
1 .48355384697159722986...   Li3 (  4) 

 Li3    
2
6
3
2  4
GR 4.373.2


log 2 x
x 2 dx
  x
 
= 8 3
x  4x
e 4
1
0
 1
.484150671381574899151... 
  (1  i 2 )   (1  i 2
2 4

(k  1)!
4 
.48430448328550912049... 

 
k
1

k 1 ( k  2 )!2

1 .4844444656045474898...

.484473073129684690242... 

3

2
18
Dingle p. 70

2  log 3 log 2 3


3
3
log 2 xdx
0 e3 x
3
64

.48451065928275476522... 


 
2
k 1
( 3k )  1

Ei (1)  
H
  ( 1) k k 
e
k!
k 1
1
log(1  x )
dx 
=  
x
e
0
.48482910699568764631... 

 
8 .4852813742385702928...
72  6 2

.485371256765386426359... 
5 2  log 2 2
 1  i
 1  i
 
 Li2 
  Li2 

 2 
 2 
48
2
4

=
1 .48539188203274355242... 
log(1  x 2 )
0 x (1  x )2
1



log 3
(1) k

  3k  1
6
( 3k  1)
6 3
k 0 2

45 2
sin 2 2 
7


 1 k (k  6) 
k 1

.485401942150387923665... 

=
x
2
2
dx
 x 1
(1) k  1
  6
k 1 k  1

.4857896527073049465...
.48600607487864246273... 
2 3
 
3


=

 log1  3( x
0


 
=
cos x
 2  cos x dx
0

1  x 3
0 log 1  x  2 dx 

1
 dx 
 1) 
2
Berndt 8.14.1

1 .486276286405273929718... 
x cos x
dx
1  sin x
0
4 G   log 2   
GR 3.791.2

=
 log(1  sin x ) dx
GR 4.224.10
0
4 .48643704003032676232... 

9 .486832980505137996...
 k  3
  ( k  2)  1 
k 
k 1

 
 ( 4)  2 (3)  1 
90 


1 .48685779828884403099... 
 (k ) (k )  1)
k 1
5 .48691648995407706207... 
.487212707001281468487... 

10 e  16 
4
  k
log 24  4 
k 1

 k !2
k 0
.48726356479168678841... 

3/ 4
2 23

.48749549439936104836... 
2 2
1 .487577370170670626342... 
1

3 .48786197086364633594... 



2

0

 (k  1)
2
1
( k  3)

2

 /4

tan x dx
0
(1) k

2
k 0 k  1 / 2



15

4
4

3/ 4 csc  14  csch  14
364 4  14

( 1) k
=  4

k  2 k  14

( k )k
.4882207515238431339...   
2k
k 1
1
3
3 log 3
.48837592811772608174... 
 2 log 2 
  x log( x  2 ) dx
4
2
0
 
.48808338775163574003... 

.48854802693053907803... 
5 1
2 2 arctan 2 
 
8
4
1 .48858801763883853368... 
 2 


 1 / 3


 /4

0

cos4 x
dx
1  sin 2 x
Hk  1  ( k )  1
1 
k

log 2


k
2 k 2
k 1
k 2

.48859459163446827723... 
Prud. 5.5.1.15
dx
3
2 2
csch
k  4

k 
4
log 3  2 2
 2 (k )
k 2
x

k
 log
.48883153138652018302... 
2  4 log 2  2 log 2 2 
2

1
 4
  1   
.488870533723461898816... 


.489461515482517598273... 
2 .48979428564930390972... 
2 .48985263014562670196... 
22 .49008835718767688354... 
k 0
2 2
2
 2

8k
)  
k  0 ( 2 k )!

2k

k  1 k ! k ( k  1)

x log 3 x
dx
3   

 2 ( 3)  
ex
2
2
0
2
3

 (k  1)
k 1
k!

eG
2
3
1 

 k  (k  1)   (k  2)  2
3
k 1
(1)
k 1
  2k 
k 1
.49023714757471304409... 
e
2
3
e2
 log 2   
 Ei ( 2 ) 
2
2

.490150432910047591725... 
2k 
 1  2k  1 
1 2
8 .4889672125599264671...  cosh 2 2 
(e
2
 3
1 .48919224881281710239...    
 5
1 .489343461075086879741... 
4
(1  x ) log (1  x )
dx
x
0
1
=
( 3)
  k4
k
13  2

 log  
16 12

k2
 (2 k )  1 

k 1 k  1
Adamchick-Srivastava 2.24
.49035775610023486497... 
1 .490664735610360715703... 

2
2

 ( 3) 

.49073850974274782075... 
40
 5
  (1)  
 2
9

2
1
 k ! ( k )
k 2
.49087385212340519351... 
5

32

=

 (x
0
dx

 1) 4
5
sin x
dx
x3
0

2
GR 3.827.9
2
dx
 (5  sin x)
=
2
0
1
=
x
3
arcsin x dx
GR 4.523.3
0
 (k )

.491284396157739513711... 
 k !k

k 2
1 .491388888888888888888 =
535 .491655524764736503049... 
k 1


1 1
 
k k
(1) k

k  0 ( 3k  2 )!


sin( x /  )
dx
ex  1
0
x cot x
  2
dx 
2
e 1
0 x 1

1
.49191653769852668982... 
6  ( 3)1/ 3    31/ 3    ( 1) 2 / 3 31/ 3 
 ( 3)  1 
2
3
1
.492504944583995834017... 
1

4
2
43 .49250925534472376576...  16e

1  e 3 

2

.492860388528006732501... 
4 2

k3  3

k3
k 2

 k  2
  ( k  1)  1
k 
k 1

 


.492900960560922053576... 
  log
 3  
1 1
  2 e cos
   
3 e
3
 2
.491714676619541377352... 
1 .492590982680769548804... 
 1
5369
 H ( 2) 6
3600
e 2   i 4 i
.491691443213327803078... 
5 .491925036856047158345... 


  Ei k  
 /4

0
cos2 x
dx
1  sin x


sin( 2 tan x )
dx
x
0

coth 2 2 
GR 3.881.2

1

16
k
k 1
2
1
8
2 2 arcsinh 1  2  2 2 log(1  2 )  2 

2
k 0
2 2 
5 .493061443340548456976... 
5 log 3
.493107418043066689162... 
.4932828155063024...

 

3

 2  6 
root of


3

0
 1
SinIntegral  
 2
3
.493414625918785664426...  
2



2 .4929299919026930579...
2
2
k
1

( 2 k  3)
2
x log xdx
ex
(1) k

2 k 1
( 2 k  1)
k  0 ( 2 k  1)! 2

Ei ( x )  1  Ei ( x )

92 


1/ 3
 cos1/ 3
2
4

 cos1/ 3
 cos1/ 3 
9
9
9
[Ramanujan] Berndt Ch. 22

 .493480220054467930942... 
2
 2 (2)   (4)

 T (2) 
20
2 (2)
1
n
2
,
where n has an odd number of prime factors
[Ramanujan] Berndt Ch. 5

.493550125985245944499... 
k
k 2
.4939394022668291491...

4
2
6 
15
x
 e e
x
0
6 .4939394022668291491...

=
( 3)
 2 


1

=
1
k2
log

2
k
4
log 3 x
dx 
x3  x2
1
log x
 x  1 dx
0
3
x
 1
dx
 6 (4)   ( 3) 1
AS 6.4.2
15
1
log 3 x
0 x  1 dx
GR 4.262.2

=
x3
0 e x  1 dx
Andrews p. 89
.49452203055156817569... 
log 3 2  (3)
 (2)  2 log 2 


3
4
.494795105503626129386... 
 2  (3k )  1
2

k

k 1
1 .495348781221220541912... 
.495466871359389861239... 

k 2
log 

1  log 2 


0
cos x
x
G8
.49559995357145358065... 
1  1 1
   , 3,  
16  4 2 
=
3
2
2
51/ 4
.495488166396944002835... 
 

k
dx
1
log 2 x
0 x 2  4 
i   i
 i 
 Li3     Li3    
 2
2   2

( 4 k  2)

k2


4
4k
k 1
k 1 4 k  1
 

 
=
 cot
 coth

8 2
2
2

1
.495691210504695050508...   k
k  1 ( 2  1) k
.49566657469328260843... 

log 2 (1  x )
0 x 2 ( x  1)
1
2 .495722117742122406049... 
3
 ( 3)
5 .495793565063314090328... 
6G

2 3
sin 4 k / 5
 
GR 1.443.1
125
k3
k 1
7 4 1 1
1/ 4
.496460978217119096806... 
    1  csc (1)1/ 4    i csc (1) 3/ 4   
720 2 4

(1) k  1
 
=  8
4
k 1 k  k
.496100426884797122808... 

.49646632594971788014... 
3 .49651490659152822551... 
.496658586741566801990... 
.496729413289805061722... 
  e  1

4e
2
4
sin 2 ( x 2 )dx
0 1  x 2

 2 (3) 
 1

2 10
2

6


11

8

Hk 2
2
k 1 k

cos 2 k
   (k  2) 
Im





k
k2
k 1
 k  1 ( 2i ) 

dx
2
5
 2x
0
31 ( 6) 7 ( 4 )  ( 2 ) 1  


.496999822715368986233... 


   coth  tanh  
32
8
2
2 4
2
2

(1) k  1
=  8
6
k 1 k  k
1
 1
 5  1
.49712077818831410991...      
1/ 4       
 4  2
 2
2

sin k

.497203709819647653589...  
 2k 
k 0
k ! 
k
.497700302470745347474... 
2 log   log 6  log ( 2 ) 

k
k 1


=
6 .497848411497744790930... 
.498015668118356042714... 
 1 

log
2 
p prime
1 p 

7 ( 3) 3 2
k 2 ( k )

 log 2  

4
8
2k
k 2

log  (2) 
Im (i )

5 .498075457733667127579... 
 (1) 
k
k 2

 (k )
2
log k
4
( k )  1 
Titchmarsh 1.1.9
HW Sec. 17.7
16k 2  6k  1

3
k  1 2 k ( 2 k  1)


2 .498215280831227494136... 

 ( k  2)
k 0
.498557794286274894758... 
.498611386672832761564... 
.49865491180167551221... 

 

 
 2k 
 
k 
k

1
k 1 2 Hk
2
2
1
(
)



log

(

)
Ei



e2
k!
k 1
k

1
1
(1)
sin
sinh
 
GR 1.413.1
2
2
k  1 ( 4 k  2 )!
1/ 3
1/ 3



1
 2 3 1   3    ( 3  3i )   H  (1) 2 / 3  3     
  2 


24 / 3311 / 6 
 2    





  3 1 / 3  
3  3i 
  4 / 3 11 / 6   H        
  2  
2 3 



k
= 
3
k  0 2k  3

3 .498861059639376104351... 
 e
k 2

1 .499005313803354632702... 
2
k 2
2 .499187287886491926402... 
e
(k )
 e
1
k 2
 (k )
G
k2 1
.49932101244307520621...   4
    ( s )  1) ds 
k  2 k log k
2


4

 .50000000000000000000
=
1

2
1

=
 (k  1)!  k !
k 1

=
  (2k )   (2k  1)
k 1

=
=
=
=
=
=
=
=
=

k  ( 2 k  1)
4k


k
2
2
4
k 1
k 1 ( 4 k  1)


1
k2



k
k
k 0 3
k 1 3



1
1
k 1





2
3
k 2 k  k
k 2 k  k
k  1 k ( k / 2  1)

1

2
k 1 4 k  1

2k 4  k 3  1

k7  k
k 2

k2  k 1

k  1 ( k  2 )!


1
1



k  0 k !( k  1)( k  3)
k  1 k !( k  2)

2k  1

2
2
k  1 ( k  1)(( k  1)  1)


1
1



3
1
3
1
k 1 k  k  k
k  2 k  3k  k

4k  1
 ( 2 k  1)!!



k  1  ( 2 k )!!  ( 2 k  1)( 2 k  2 )

=
2k

k  0 k !( k  2)( k  3)

=
 (1)k  (k )  1 
k 2

=

=
=
=
 (2k )
k


  (2k )   (2k  1)
k 1
k ( 2 k  1)
4k
k 1


4
(k )

k
k 1 2  1

 ( 2 k )2 k

k
k 1 4  1
2
3



k 1 sin k
k 1 sin k
k 1 sin k
(

1
)

(

1
)

(

1
)



k
k2
k3
k 1
k 1
k 1
k 1

GR 0.142
K 134
2

=
J397, K Ex. 109d
J390
8 .525161361065414300166... 
.525200397399770342589... 
log 7!
 ( 4 )   ( 3)   ( 2 )  1 

k
k 1
5
1

 k4

=
1
  (1) k 1  ( k  4 )  1
2 k 1

35/ 6
35/ 6
e3 / 2 
4/3
3

sin

cos
 e3 / 2  
5/ 3
2
2
3


1/ 3
.525223891622454569896... 
1 .52569812813416969091... 
 3/ 2
2
coth  3 / 2 
3  1

2(  1)

 ( 1)
k 1
k 1
2
1
3
 arctan 2 
 csc
 sin 

5
5 5 2
 2 
1 
.525738554967177620202... 
3

 (k )
3 .525956365299825031078...  

k  1 ( k  1)!
.52573111211913360602... 

2 .52605639716194462560... 
 4 log 2
64
3 2 (3)

 93 (5)
16
 /2
=

x
3
log cos x dx
0
.526207036980384918178... 
 ( 3)
 ( 3)   ( 4 )
.526315789473684210
10
19
=
(1) k
.52641224653331...
 
k  2 k log k

Hk
.52693136530773532050...   k 4
k 1 2 k

(1) k
.526949261447891738811...   5
k 0 k  1

.526999672990696468620... 
3 .527000471852952829762... 
.52706165278494586199... 
5
27
log

16
5

 1 (k )
k 1
k!

 arctan
1
1 e

sin5 x
0 x 2 dx

3k k


k  1 ( 3k )!
 k  (2 k )  1
7

3
1 .527525231651946668863... 
.527600797260494257440... 
.527694767240394589488... 
 2k  1

  k  7
k 0
k
1
 (  1)(1  cos1)  log(2  2 cos1) sin1 
2
 2
7

12
coth  2 
4

k
k 2
2
1

2

 ( 1)

k 1
k 1
k 1
sin k
 k (k  1)
2 k 1  (2 k )  1 

=
.527887014709683857297... 
1


2
k 1 k  2k  3
k 1
( k  12 )!
( k  12 )!( 2 k  12 )


 
=

k 1
1 .52806857261482370575... 
log  
2  (2k )  1
2  
k

2 csc 
k
k 1

2

   log1  2  
 k 
k 2
.528320833573718727151... 
log8 3
.528407192107340935857... 
 6  37 / 6 i  32 / 3 
1   6  37 / 6 i  32 / 3 
1 

 
  
    1  1/ 3     

3 
6
6
3 





( 1) k 1

 k ( k  1 / 2) 
  4  2 log 2 
 
=

 ( 3k  1)
k
3
k 1
.528482235314230713618... 
2
4

e


1
 k (3k
k 1

2
k 0
.529154857165146540082... 
.5294117647058823
4
40
9
=
17

.529416855888971505585... 
e
 x
dx
0


 2 log 2 2
1
3k
k!
 k ! 1 
k 2

 1  dx
2
 x
 sinh x
1
1
1/ 2
F
k 1
2 .52947747207915264818... 
k
 1)
1
1
.52862543768786425729... 
SinhIntegral (1) 
2
.52870968946993108979...   G
1 .528999560696888418383... 
3
3

2 log 4 2
 4 ( 3) log 2 
3

 2k 
  k  4
k 1
1
k4
k
. 52962384375569377748... 



5 .529955849000091385901... 
2
 2k 
1

   
k
k  0  k  16 ( 2k  1)( 2k  2)
4G  2

k
k 

  1  2
k 1

(1) k 1
 (k  1)  1 

k
k 1
1 1
.53019091763558444752...   ( 3)   
  (2  i)   (2  i) 
2 2


1
1
 
=  5


( 1) k 1  (2 k  3)  1

3
2 k 1
k 1 k  k

.53001937835233379231... 

k 1
k
k 2

1
.530277182813685398637...  
k  1 ( 2 k )! k

.53026303220872523716... 
1
k
2
2
4 k 2  3k  1
1 .530688394225490738618... 
 2 ( 3)  1   (1) k  ( k )  1  
3
2
k 2
k  2 k ( k  1)

1
1 .53091503904237871421...  
Titchmarsh 14.32.3
k  1 k ! ( 2 k  1)
9488 .531016070574007128576... 

 .531249534072353755050... 


k
2
8
1 
 1


 
6
 3



 log1  6 x
1
2
0

 dx
 1
7
k
 (1) k
k 1

k 1  ( k )
.531275083024229991483...   ( 1)
log (2 k )
k
k 1

k
1/ e  1
.531463605386615672817...  e
 
k
k 1 k ! e
.531250000000000000000 =
2 .531895752688993200213... 
.531972647421808261859... 
.53202116874184233687... 
.532108050640355849704... 
17

32
 e
33/ 4 

2 2
x
2
6 
3
8


0
Prud. 5.3.1.1
k
4
dx
1/ 3

 2k 
  k  12
k 0
k
1
( k  1)( k  2 )
1  1 3
 Li3      (3) 
12 2  3 8
2


1
4
  2 log 1  2
2

1
log 2 x
0 ( x  1) 2 ( x  3) dx
(1) k 1
 

k 1 k  1 / 4

1
2 .532131755504016671197... 
.532598899727660345291... 
e
2 I 0 (1) 
3
2
4
 sin  x
1

dx
3
 k (k  2)

k 1

2
 /4

2  log 1  2  
.53283997535355202357... 
 cos x log(sin 2x ) dx
0
7e  1

3
3
46 .53291948743789139550... 
.533290128624160141531... 

 
=

1
3
k
3 k
2
 (k  1)!
k 1
 3  i 3
 3  i 3
1
  (1  i 3 )  
 
  (1  i 3 )  
6 6
 2 
 2  

2

1
  ( 1) k 1  (3k  2)  1 
2 k 1

1 .533463799145879760767... 

  
k 2
( k ) 1

k
k 1
5
1
 k2
 1

k4
6 .533473364460804592338...  62 e  162  
k  1 k !( k  3)
3 (3)
1 .53348137319375099599...   2 Li3  2  

2

log 2 x
 
= 
dx
x

1
x

2
(
)(
)
1
1 .533582691060658478950... 

1
I 0 (1)  L0 (1)  1   e x arccos x dx
2
0
1 .53362629276374222515... 
e


2 ( k )  1

( k )
k 2
1
3
 1
.53387676440805699500... 
 sin  2 sin  
4 cos1  5  2
2
(
)

k


|  (k ) |
2
1 .53388564147438066683...   k
  k
k 1 2  1
k 1 2
 1
7 .533941598797611904699...     
 8

.5338450130794810532...





k 
(1) 

.534195237508799281491...   1 
  2k  
k 1
  k 
  k  


1
2
k 1
k
sin
2k  1
2
3 .534291735288517393270... 
.534304386409498279676... 
.53464318757261872264... 
1 .534680913814755890897... 
.534799996739570370524... 

Hk  1
k 
k 1 4


1 1 1
1
  , 2,    k

2
8  2 2
k 1 2 ( 4k  4k  1)

1 e / 4 1/ 4 e
cosh k
e e   

k
2
k  0 k !4
1 

log 1 
 

2
 /4

log x
dx
2
1
 2x
1
cos x
 1  sin x dx
0
1  1 i
1 i
 1 i
  1  i 
 
 
 
 
 2
 2 
 2 
4  2 


1
1
k 1
 
=
  ( 1)  (4 k  1)  1   5
2 k 1
k 1 k  k
 sin 1 
 cos1
 cos1 
 sin 1
.535112163829819510520...  sin
  cosh
  sinh
   e(cos 1) / 2 sin

 2 
 2 
 2 
 2 

sin k
= 
GR 1.449.1
k
k 1 k ! 2
.535034887349803758539... 

9
8
16
4
log  1 
3
3
.5351307235543852976...

   
3

sin

.535216927240908236283... 


1 

  1  2 
3k 
3
k 1 
1
 k !(k ! 1)
k 2

1 .53537050883625298503... 
1
F
k 1

 ( 3) 

Ai(1)
.5355608832923521188...
k 1
2k
1 .535390236492927618733... 
 2k

5
 4k  1

 1
1 

5   2 k
 4k
 1   1 
k 1  

1
3


2( 2  3 )  3  1 2

1
.535962843190222987031... 
  ( 1) k 1  (5k )  1 
2 k 1
.535898384862245412945... 
.5360774649700956698...

.536119444744722773227... 
1 .53642191914104435977... 
( 2  1)   1 
  
 2
2

e
0
k
k 1
1

1
5
dx
x2
1

 e
 2


CFG D1

3
csc 2 


 1  k
k 1
2
1


 4k  2 


1
.536441086350966020557... 
  ( 1) k 1  (6k  1)  1 
2 k 1


1
 2 ( k )
.5365207790738756208...   k



2
3k
k 1 (3  1)k
k 1

k
k 1
6
k

1

.536526945921177100962... 
 log  ( x) dx
2
.536683562016736660896... 
.536784534109508703918... 
1

2

 (1)
k 1
=
k2
 (7k  2)  1   k 7  1
k 1
k 1
 (1)  (3k  1)   (3k  1)   k (k  k  1)
k 1
k 1

.53685577365418220547... 

 1  i 3
 1  i 3 
1
 (1  (1) 2 / 3 ) 
  (1  (1)1/ 3 ) 
   
3
 2 
 2  

 
k 1

2
k 2

  ( jk  2)  1
j 1 k 1

.536921592069942010965... 
4
k 1
k
k
1
1
2

  Re (1)  i 
2 2 sinh2 
7 (3) log 3 2  2 log 2
 1
.537213193608040200941... 


 Li3   
 2
8
6
12

H ( 2) k  1

 
 
k3
k 1
.536999903377236213702... 

2
k 1
1

k3
k
log(1  x / 2) log x
dx
x
0
1

 
=
1 .53739220806790406178... 
.538011176205005048612... 

 
=

 1/ 3 3
1  1 / 3 2   ( 1 / 3 ) / 2
e
cos
 e

3
3
2

k
 (3k )!
k 0
5
1 5 
1
log
 k

2
2
k 1 5 (8k  4)
1  (1) k 1 (2k  2)!!

2 k 1 (2k  1)!!2 2 k 1
H ( 3)1 / 4  64  27 (3)   3

1
1 .538395665719128167672...   sin
k!
k 1
7
.538461538461538461
=
13
.5381869343911341187...

J141
1 .538477802727944253157... 
.539114847327011158805... 

2 coslog 2  2i  2  i 
1
 1  i
 1  i
 1  i
  tan
  4 csc
 
 i tanh
 4 
 4 
 2 
8
1
1 i
1 i
 1 i
  4i csch 
  cot 
  i coth
 
 2 
 4 
 4 
8
 


 
=
e x cos x
 1  e 2x dx


 (1 / 2 )
( 1) k

J1028
 1 
 ( 5 / 4 ) (3 / 4 ) k 1
2k
 4  i 21/ 335/ 6  61/ 3 
 4  i 21/ 335/ 6  61/ 3  
1 
1/ 3
2/3

3 .539356459551305228672... 
1
1
(

)




(

)


 
32 / 3 
2
2




.53935260118837935667... 

 

21/ 3
1/ 3
  2 / 3   2  6  
3
=
 6  (3k )  1

k

k 1


91
8 .539734222673567065464... 
 e
9 .5393920141694564915...

k
k 2
Not known to be transcendental


.540235927524947682724... 
 3 log
.540302305868139717401... 
cos1  cosh i  Re e 
.540345545918092079459... 
.540553369576224517937... 
.540722946704618254025... 
3 1
i

4

2
8
cosh
6k
6
3

4

( 1) k
 
k  0 ( 2 k )!
1

2
 log 2 
1  4k  1


2 k 1 ( 2 k  1) 2 k
(1) k  1 sin 2 k

k !2k
k 1

 2k  k 3
37
6 .540737725975564600708... 
   k
4 2
k 0 k  8
5 .540829024068169345604...   ( ( 3))
 
=
.540946264299396859964... 
G7 
1
8 .54097291002346216562... 
log 3 x
3 ( 2 )  3 ( 3)  
( x  1) 3
0
AS 4.3.66, LY 6.110
( 1) k k  ( k )
2k
k 2



 cos 2 
 cos 2  
sin(cos1 sin 1) cosh
  sinh
 
 2 
 2 





k6
45 .54097747674216511051...  13I 0 ( 2 )  10I1 ( 2 )  
2
k  1 ( k !)
.541161616855569095758... 
.5411961001461969844...
4
log(1  x ) 2 log x

 
180
x
0
2 2
(1) k 

  

 cos  sin   1 
2
8
8 k 1  4k  2 

1
.54123573432867053015... 
1
J1029
dx
 ( x)
0
(1) k Bk
.54132485461291810898...  log(e  1)  

k !k
k 1

Bk
.541608842204663463822...  
2
k  0 ( k !)

1
1 .541691468254016048742...   k
k  0  2 !

1 .541810518781157499421... 
=
2 .541879647671606498398... 
.542029902798836695773... 
178 .542129333754749704054... 
.542720820636303500935... 
1 .54305583321647144517... 
log 3

4 3
1
 2  26 ( 3)



  ( 1)   
 3
2
27
6 3 81 3 3


36k 2  9 k  1
k 2 ( k )
 

3
3k
k 2
k  1 3k (( 3k  1)


1
( k  1) ( k )
(1)  3 
2
  8G      
2  
 4
4k 2
k  0 ( k  3 / 4)
k 2
2
cosh 

H ( k  1( k  2 )( k  3)
112  96 log 2   k
2k
k 1

1
1
1
sinh
 
k
2
2
k  1 ( 2 k  1)! 2

p prime
.543080530983284218602... 
=
[Ramanjuan] Berndt Ch. 5
p
p!
i   i    i 
coth          

2
4
2 8   2   2



2
i
i
i 


   ( 1 )  1     ( 1)  1   


8
2
2 

Hk

2
k 1 4 k  1
2

 


.543080634815243778478... 
1 .54308063481524377848... 

  1  
k 0
.543195529534402361791... 

 
=
.54323539510879497886... 
e  e 1  1

2
k  0 ( 2k )!
cosh 1  cos i 

=
e  e 1  2  1
 1  dx

  sinh   2
2
 xx
k 1 ( 2k )!
1
cosh 11 
2
GR 0.245.5
4

2
( 2 k  1) 
J1079
1
log(1  ei )  e2i log(1  e i )(sin 1  i cos1)

2

sin k

k 1 k  1
  ( k ) log  ( 2k )


k
k 1

.543258965342976706953... 
1  5  1
     
 3  6  6
.543383238748395175193... 
4
8 .5440037453175311679...
2
6
, Landau constant
 4 log 2  2 log 2 2 

Hk
 (k  1)(2k  3) 
k 1
73

2



 2  i 2  i     coth  tanh 
sinh 
2
2


log  (2k )
.54425500107911929426...  
k
k 1
.544058109964266325950... 
.54439652257590053263... 
 log 2
4
 
=

0
 /4

 
=


0
 
 log sin(4x ) dx 
0


 /4
sin x
1  sin 2 x
log sin x dx 
 
 dx

   x  tan x  cot x 


 2
 cos 2 x

.54445873964813266061... 
  ( k ) (2 k )  1
k 1
1 .545177444479562475338... 
8 log 2  4 

 k (k
k 1
.545454545454545454545... 
6

11

Lk
4
k 1
k
2
1

 1 / 4)
Prud. 2.5.29.28
.5456413607650470421...

.54588182553060244232... 
1
e1 / 4  
1  erf  
2 
2
7 4
 9 (3) 
60
  2 arctan
1 .5459306102231431605...

.546250624110635744578... 
2 2

5
tan
2
 
=

ex
e
x2
dx
1

log 4 x
1 ( x  1) 3 dx 


x
2
0
 5

 (1)
k 2
k
1
x log 4 x
0 ( x  1) 3 dx
dx
 2x  3
Fk 1  (k )  1 
2 (1  5 )


1
2

 2k  1  5 k  2  k 5 
k 2
1 .546250624110635744578... 


 
=

1
5
k
k 2
tan
 5
2
1

2
 k 1


 F  (k )  1 
k 2
k 1
2 (1  5 )

 2k  1  5 2  k  k 5 
k 1
1
.54716757473605860234... 
.54770133168797308275... 

 
=
1  log 3  log(e  2 ) 
ex
0 e x  e / 2 dx

1 1 i 1 1 i
 log 2           
2
4  2 2 4  2 2

1

2
k  0 ( 2k  1)( 2k  1)  1
Prud. 5.1.26.8
arccos 2 xdx
arctan 2 x
7 (3)
1 .54798240215774230466...  2 G 


4
0 x dx 
2
1 x2
0
1
 /2

 
=

0
.54831135561607547882... 

 
=

 
=
11 .548739357257748378...

.54886543055424064622... 
.54891491425246963638... 
2
x2
dx
sin x
( k  1)!( k  1)!

18
(2 k )!
k 1
1
2 arcsin 2 
2

log(1  x 3 )

dx 
x
0
i sin(2 log i ) 

3 1
(1) k 1 (k!) 2

2
,
2
,

F
,




2 1
2 4  k 1 (2k  1)!


1

k
k
k 1 2 ( 2  1) k



1
  ( 0)
1   (1)
1

( 0)
2  (1)
log 2   1 
.549306144334054845697... 
log 3
1
 arctanh 
2
2

=
 
=
2

dx
0 (x  1)(x  3) 



.54926923395857905326... 
log x
 ( x  2)
2


2
k 0
2 k 1
dx
2 x 2  1 
1
(2 k  1)

e
0
Titchmarsh 2.12.8
AS 4.5.64, J941
dx

2
x

1
.549428148719873179229... 

 

1
csc ( 1)1/ 4   csc ( 1) 3/ 4   (cos( 2  cosh( 2 
4
3/ 4
1/ 4
1/ 4
3/ 4
  ( 1)  sin ( 1)    ( 1)  sin ( 1)  ) 
(1) k

 
=  4
k 0 k  1
57 
7 .5498344352707496972... 
1
8 8
log 7 x
( 1)


dx
5060 .549875237639470468574... 
 7! (8) 
(7)   

x
15
1
0

GR 4.266.2

2 .550166236549955...

 (1)
k 1
k 1
3
1 .550313834014991008774... 
6 .550464382223436608725... 
.550510257216821901803... 
e1/ k
k3

20
6
G
3 6

.550521946799569348190... 
  (k  1)  (k )  1
k 2
6
1

  6  ( 2)  1
1
1
.55066059449645688395... 
 coth  2 
2
2
2
2

 (k )
.550828461280021051067...  

k
k 1 2
1 .550546096730430440287... 
 2 2
.55154451810789954008... 

k 0
 
=

2
1
 2 )
4
1
 2  5  2 H ( 3
10 


 2

 (k
 2 (1   )  (2)   (2) 
6
3
36

x log s x
 
=  x
dx
e

1
0
4
2
1
.551441129543566415517... 

2
2 
e e
sinh 2
sinh 1 cosh 1
2 .551419182892557505235... 

2
F
k 2
k 1


  1  5 tan
5) / 2
Fk  (k )  1
13  k 2
.5516151617923785687...  e 


6 k 1 (k  1)!
.5516932...


p prime
.5523689696799021586...


8
1

p 1
2
erf 1  erfi1


 1  dx

4
 x3
 cosh x
1


.55242828301478027042... 

1
1 .55260145083523513225... 
8

 ( x  1)
x 3  x 3
cos
dx
 21
2



k 1

1

 1  k (k  5) 

 5 

2 
J132, J713
.5527568077303040748...

sin 


1 
2

 1   k
k 1
1

5
.55278640450004206072...  1 


( 1) k 1 ( 2 k )!

( k !) 2
k 1

( 1) k 2 k
.55285569203859119314... 
2 sin 2  cos 2  1  
k  0 ( 2 k )!( k  1)

1
.553303299720515717371... 

2 arcsinh 1  log 2   k
k 1 2 k ( 2k  1)

 0 (k )
2 .553310333104003190087...  
k  2 k ! 1

.5535743588970452515...

.554204472098099291561... 
arctan 2

2

x
2
0
dx

 2x  5
i  (1 )  2  i 
 2  i
 4  i
 4  i
   ( 1) 
   (1 ) 
   ( 1) 
 
 
 4 
 4 
 4 
 4 
8
( 1) k 1 k

2
2
k 1 ( k  1 / 4)

 2k  1
   
k 
k  1  k  k !! 2


 
4 .5544893692544693575...
=
1 .55484419730133345731... 

1
cosh

3
2


2
 1  (2k  1)
k 1

2

 



1 .555290108843124661542... 
Hk
Fk
k 1
.55536036726979578088... 

2
k

dx
 2

4 2 0 x 8
 /2

cos3 x sin x dx 
0
1 sin 1  cos1
cos x dx



2
2e
ex
0
1
.5553968826533496289...

e

 
=
coslog x
dx 
x2
1


1
.55582269181411744686... 
I 0 (1)  Struve0 (1)   e sin  x dx
0
.5560460564528891354...

 2
log 2
2
2 2



log
3
6
2 2
6
1
i
i
.55663264611852264179...    Li4 (  e )  Li4 (  e )  
2
1
x
0
2
1

log 1  4  dx 

x 


 
=
 (1)
k 1
k 1
.55683841210516333558... 
 
=
F
k 1
k 2
.55730495911103659264... 

GR 1.443.6
1
 2  5  2 H ( 3
5 


cos k
k4
2
tan 1 
6 .55743852430200065234... 
43 
 5 

2 
Fk 1  (k )  1
1

log 2
1 .557407724654902230507... 
  tan
5) / 2

2
k 1
1
k
1  1 / 2 
1/ 2 k
(1) k 4k (1  4k ) B2 k


( 2k )
k 0

4 log 2  log 3
H2 k

k
3
k 1 4
1
 log 6 e
.558110626551247253717... 
log 6

H
.5582373008332086382...   k k 3
k 1 2 k

.5579921445238905154...

1 .558414187408610494023... 

1  cosh 2  2 sinh 2 

.558542510602814021154... 
J153


2k

k  0 ( 2 k )!( k  1)
 ( k  1) log k
2k

log k
1 .558670436425389042810...  

k  2 ( k  1)!
k 1
.559134144418979917488... 
162 .559234176474304999069... 

1
log n k

 k!
n  1 ( n  1)! k  2
(1) k
J0 ( 2 )  
( k !) 2 2 k
22 e2 

2k k 3

k 1 k !
  (k )


 1 

k  2   ( k  1)


.559334724804927427919... 


k
k 2
 (k )
2
(k  1)

e
1
 1 
erf 
  
2
 2
k 1 k!!( k  2)

7
(1) k 1 3k
 3
 3
.559615787935422686271...  log
 Li1     Li1     
4
4k k
7
 4
k 1
.55940740534257614454... 
e
5

2
3 .55989038838791143471... 
27 
.560000000000000000000 =
14

25
39 (3)

2

 (1)
k 1

.560045520492075839181... 
k 1
 ( 1) k 1
( 1) k 1 




3
( k  1 / 3) 3 
k  1  ( k  1 / 3)

k 2 Fk

2k

x sin 2 x
0 ex dx
1
 k !2  (2 k  1)
k 1

Titchmarsh 14.32.3
k
1

k 

k 1

1
1

k
= 1   (1) 
 ( 3k 2  k ) / 2 
( 3k 2  k ) / 2
3
3

k 1




k

 (1) 
.560329854485890998637...   1 
 2k  
k 1
  
  k 
  
.56012607792794894497... 

 1  3

.560511825774805757653... 
9 e1/ 3  12 
Hall Thm. 4.1.3
1
 k !3 (k  1(k  2)
k 0
k


10
 (k )  1
2
.560744611093552095664... 
 4 log 2   ( 1) k


k 2
3
2
k 2
k  2 k ( 2 k  1)

 0 (k )
1 .560910575045920426397...  
2
k  1 ( k !)

2 3
1 ( 2)  1 
1
.561061199700803776228... 
13 (3)  27       27  
3
2
3 3
 3
k 1 ( k  1 / 3)

1 ( 2)  1
1
27 .56106119970080377623...       
1 3
 3
2
k  0 (k  3)
1 .561257911504962567051...  4 ( 3)  3 ( 4 )
(1) k  k

Euler-Mascheroni constant
k!
k 0

 1  1 / k

KGP ex. 6.69
=  1   e
k
k 1 
 (n) log log n
= lim

HW Thm. 328
n


1 

 
= lim  log n 
Mertens
1



p
n  
p prime  n 


log 4 xdx
3 2
4
2 2
     

 8  (3)  
20
ex
0

.56145948356688516982... 


 
23 .5614740840256044961...
e   

.56173205239479373631... 
1
e  ecos 2 cos(sin 2)  1 
2
.561781717266978021561... 


6
sec
 5
2
 log 2
.562500000000000000000 =


 1  k
k 3
9 (3)


16
3
.56227926848469324079... 

8
2

cos2 k

k!
k 1
1


 k  1
1
arcsin 3 x
0 x dx 
 /2

0
9
16
2k 2  1
=  k  ( 2 k  1)  1  
2
2
k 2
k  2 k ( k  1)

( 1) k 1 cos k
1
.56256294011622259263... 
log2  2 cos1  
2
k
k 1
.562610833137088244956...   ( 2 )   ( 4 ) 


1
1 

.56264005857240015056... 
sin
  1  2  
2
2 k 1  4k  1 



1 .562796019827922754022... 
 (1) 
k 1
k 1
4 .5628229009806368496...

2  log 2 

5 .5632758932916066715...

x3
dx
tan x

k 1
2
J1064
( 2 k )  1

3
log( 3  2 2 ) 
2 2
kH2 k
k
k 1 2

pd ( k )

2k

k
1




k  0 k!( k  4)
k  0 k!( k  3)

1
=

k  0 ( k  1)!  3k !

1  (1)  1 
( 1) k
(1)  2  




8 .5636594207477353768... 
 
 


1 2
 6
 3 
4
k 0 (k  3)
1
1
cosh 
.563812982603190392613... 
2
2

 ( 3) 3
6k 2
1 .56386096544105634313...  6 ( 2 )  6 ( 4 ) 
  
4
3
8
k  2 ( k  1)
.5634363430819095293...

6  2e 
=
 ( 1)

k 2
.56418958354775628695... 
10081 .56420457498488257411... 
1


17 8
k

( k  1) k ( k  1)( ( k )  1)

 (k 
k 0

k
1
2
)! 2 k
x7
 
dx
16
sinh x
0
GR 3.523.10
2 .564376988688260377856... 
 1
 2

.564599706384424320593... 
 (1)k
k 2

.56469423393426890215... 
 (k )  1

1  1
  log1   
k 1
 k
k 2 k
sin k
 
k
2
erfi 2 

1 .56493401856701153794... 


1
  1  3 
k
k 1
1 .564940517815879282638... 
1 .565084580073287316585... 
.565098600639974693306... 

 


 
=
Berndt 8.6.1
  2k 
k 1
18 .564802414575552598704... 
 1
 4
    3    2  
2 2 k 1

k  0 k !( 2 k  1)

Q( k )
 1  k
k 1 3



tanh  
2
sin 2 x
 sinh x dx
0
1/ 4
6
1
1
1




e
e
e
1
1
1
1
1
1
   e   cosh int 2   e   sinhint 2   e   log 2 
2
e
2
e
2
e


  e     cosh 1   e   cosh int 1   e   sinh int 1 

Hk
 (2k )!
k 1
.565159103992485027208... 
I1 (1)

11 2
1  1 i 
cos  k / 4
 1  i 
  Li2 
   
  Li2 
k2
192
2  2 
 2 
k 1
1
i

Li3  ei  Li3  e  i 
1   2  Li3  ei 
2
12

k 1 cos k
=  ( 1)
k3
k 1
.565446085479077837537... 
.5655658753012601556...

 
.565623554314631616419... 
  
2 arctan 2 



 /4
4


0


 
cos2 x
dx
1  sin 2 x
( 1) k 1 64 k

(4 k )!
k 0
6k

k 1 2
2 .565625835315743374058...  1  cos 2 cosh 2   ( 1)
( 4 k )!
k 1
1 .565625835315743374058... 
 cos 2 cosh 2 

GR 1.413.2

2 .565761892331577699921... 
k 2 Hk
e  1   ( e  1)  eEi (1)  Ei (1)  
k  1 ( k  1)!
 2 log 3 log 3 3 2  1
2
1 .56586036972271770520...   Li3 (  3) 

 Li3    
3
9
9
3  3


 

log 2 x
= 8 3

x  3x
1
. 56588424210451674940... 
1 .566082929756350537292... 


  (1)
2 J1 2 2 

I0 ( 2 ) 

2k k
( k !) 2
k 1
k 1

log 2


8
4

.566213645893642941755... 
2
 2k 
1
 


k
k  0  k  16 ( k  1)( k  2)

16

9
.565959973761085005603... 
.565985876838710482162... 
x 2 dx
0 e x  3 
1
 (4k  1)(4k  2)
k 0

1
 (k !)
k 0
2
2k
 (k )
3k
 1
5 .566316001780235204250...    
 6
12 .566370614359172953851...  4 
k 1
log(1  x )
dx 
e x 1
0
1
.5665644010044528364...

e ( Ei ( 2)  Ei ( 1))  log 2 
2


2
sinh  2

9 .5667536626468872669... 
  1  2 
k 
 2
k 1 

k!
2 .566766565764270426298...  
3
k  0 ( k !!)
15 
3
.566797099699373515726...  log
 (1   ) 
8
2
arccot
.567209351351013708132... 
3  6 log

.567384114877028322541... 
1 .567468255774053074863... 
.567514220947867313537... 
k 2
 (k )  1  3
k
k
  
 2
2
2

3

1
 3 (k  1)(k  2)
k 0
k
1
 (2k )
k 1
 (1)
k
Srivastava
J. Math. Anal. & Applics. 134 (1988) 129-140

.566911504941009405083... 

k
 ( e)
log    
 (k )

1 
1



  2 log 1   


k 1
 2k  
k
k 2 2
k 1  k

.567667641618306345947... 
5 .56776436283002192211... 
.567783854352055695476... 
1
1
 2 
2 2e
31
2 ( 3) 

 (1)
k 1
k 1
 2 log 2
2k
( k  1)!
4 log 3 2


3
3

 2k 
  k  4
k 1
1

k3
k

(1) k log k

k  2 ( k  1)( 2 k  3)

.568074442278171...

.568260019379645262338... 

 
=
5 .568327996831707845285... 
.568389930078412320021... 
.568417037461477055706... 
2


coth  

1

2
k
4
1

 k2
6
2
k 1


1
1

  (1) k  1  2 k  2 )  1   Im 2

2 k 1
 k 1 k (k  i ) 
 3/ 2 
1

2
(k )

k
k 2 2




1
 2
k  2 j 1
3 
3
erf 1 

4
2e
kj
1
e
 x2/3
dx
0
(1) k
4  J 2 k 1 (1)



 k  0 (2k  1)!!2  k  0 2k  1

k 1  (k )
.568685634605209962010...   ( 1)
2k
k 1
.568656627048287950986... 
1 .56903485300374228508... 
.569356862306854203563... 
H 0 (1) 
2

e
 6  37 / 6 i  32 / 3 
 6  37 / 6 i  32 / 3  
1 
1/ 3
2/3

(

)

(
)

1
1





 
34 / 3 
6
6




1 
1 
4 / 3   1  1/ 3 

3
3 


 (3k )
1
= 


3
k
3
k 1
k  1 3k  1
.569451751594934318891...  4 ( 2 )  5 ( 3)


1 .56956274263568157387... 
 (sinh k ) (2k )  1 
k 1
.5699609930945328064...


  (2)
1  log k



 (2)
 (2) k  2 k 2
log p
2
p prime p  1

Berndt Ch. 5
 (k )
2
k 1 k
 1
Ei      log 2 
 2

=
.57015142052158602873... 


2
k 1
k
1
k !k

H ok
3
1
arctanh
  k
2
3
k 1 3
.570490216051450095228...   cosh cos1  sinh cos1cos1sin sin 1  sin 1(cosh cos1  sinh cos1  cossin 1) 
.57025949722570976065... 


 
=
sin k
 (k  1)!
k 1
2 .57056955072903255045... 
.57079632679489661923... 

 2 (k )
k 1
2k


1 
2

sin 2 k
k
k 1

 2k 
1

  k

k  1  k  4 ( 2 k  1)

=
(2 k  1)!!

 (2k )! (2 k  1)
J388
k 1
( 1) k 1

2
k 1 2 k  1 / 2
1
arccos x
= 
dx
(1  x ) 2
0

=

=

0
tanh x
dx
ex

=
 (1  x
0

=

x dx
) sinh  x / 2
GR 3.522.7
x log(1  x ) dx
0
1 .57079632679489661923... 
2

2
GR 4.292.2
1  x2
  i log i
( 1) k

k 0 k  1 / 2

1

= 
2
k  0 ( 2k  1)  1 / 4

=

=

(1) k 1 k 2
 k
k 1
2
 1/ 4

2
k!
 (2 k  1)!!
K144
k 0

=
2k


k 1  2 k 
 k
k

( k !) 2 2 k


k  0 ( 2 k  1)!

 2k 
  k  4
k 0
k
1
(2 k  1)

=
( 2 k )!!
 (2 k  1)!! k 2
k 1
k



 
=
  Lik (i )  Lik (i )  
k 1


=
2
 arctan  (2k  1)
2







1

=
arctan 

2 
k 1
 (2k  1  5 ) 

4k 2
= 
k 1 ( 2 k  1)( 2 k  1)


dx
dx
=  2
  2

( x  1) 2
x 1
0

k 0

=
x
2
0
[Ramanujan] Berndt Ch. 2
Wallis’s product

e
0
x
dx
 2e  x  2
dx
 x 1/ 2

=
[Ramanujan] Berndt Ch. 2
sin x
0 x dx 

sin 2 x
0 x 2 dx

=
x2  1
0 (1  x 2 ) 2 log x dx
GR 4.234.4
 /2
=
 cos
2
x dx
GR 3.631.20
0

=
1
 dx
x
Prud. 2.5.29.10
 log(e tan x) dx
GR 4.227.3
  x  cot x 
0

=
arccosh x
x2
1

 /2
=
0

=
e
0

=
x
1
 2e  x  2

1
4  dx

 x log1  x
0

=
 ci ( x) log x dx
0
2 .570796326794896619231... 


2k
1  
2
k 1  2k 
 
k
GR 6.264
3 .570796326794896619231... 
.571428571428571428
=


2k
2  

2
k  0  2k 
 
k
4

7
1  p 4

2
 p4
p prime 1  p

1 .57163412158236360955... 
1


2
3/ 2
k 2 k  k

.57171288872391284966... 
(1)
k 1
 12k  2

k 0

  k / 2  1
k 4

12

1
3 1
 log
2
3 1

1  1 
 1 
.571791629712009706036...      
 
  

2   3
3 

1
= 
2
k  1 k ( 3k  1)
3 .571815532109087142149... 
9 (3) 111


4
128
4
4
 1  log  
9
3

 ( 2 k  1)
k 1
3k


 k  (2k  1)  1
3
k 2

kHk
k 
k 1 4
log 
1
 log  
.572364942924700087071... 
2
 2

2 x
e
1  dx
=   
 x

2
e  1 x
0
.572303143311902634417... 
.572467033424113218236... 






1
2

1

4


 k  (2k )   (2k  1) 
12
k 1
1

=
 Li ( ei )  Li2 ( e i )
2 2

k

=  3

2
k 2 k  k  k  1


k   ( k )   ( k  1)

=  ( 1) k 
 1


2
k 2

k  1 cos k
=  ( 1)
k2
k 1

H
2 3
.57246703342411321824... 
   2 k 
12 4
k 1 k  1
.57289839112388295085... 
4 2
8
sin 4  x 2 
  
dx
x2
0


GR 3.427.5
3 ( 3)
  ( 3)   ( 2 )
2

1 

2 .57408909729511984405...    1 

( k !) 2 
k 1 
.57393989404675551338... 
.574137740053329817241... 
.574859404114557591023... 

 
=
5 .57494152476088062397... 
2

 1


cos2 k

k2
k 1
6
2
1  2 1
   (1)1/ 3      (1) 2 / 3  
3
3

1
  (1) k 1  ( 3k  1)  1
2 k 1

e e 1 
 k

e
k 1

1/ k !
k 1
4
1

k
.575222039230620284612... 
10  3
 /2
.57525421205443264457... 

0
.57527071983288752368... 
x sin x
dx
2  cos2 x
2
3 (3)
 3 log 2 

8
4
2
log 3 (1  x )
0 x 3 dx
1
1
log 2 x
1 .575317162084868575203...  16  12 ( 3)  
dx
x
0 1

1
.575563616497977704066...  1 

1/ 4 erfi
2e
2

 3 
1 .57570459714985838481...  log      
 4

.575951309944557165803... 
2
 
2
(1) k k !


k  0 ( 2 k )!

1
i
 (1  i )   (1  i )   (1) (1  i )   (1) (1  i )

2
4


 
=
 ( 1) k  ( k )   (2k  1)
k
k 2
3
.576247064560645225676... 
.576674047468581174134... 
  4
32

2
 
=
k 0
k
2
1

1
x
6 sin  5

 5
=
k 1

  (4k  2)   (4 k )
J124
k 1
( 1) k


k  0 k !( k  1)!
.57694642787663931446... 
=
 (1)  (2k )  1 

J1 (2) 
=
Davis 3.37
sin x
x
 1
.576724807756873387202... 
=
cos( 2 k  1)
( 2 k  1) 3
 e e
0
.57721566490153286061... 

k
k 1
k 2

=

 (1)
coth   1 





k 1

( 1) k 1 k

2
k  0 ( k !)

1

 1  k (k  4) 

 (Euler's constant, not known to be irrational)
lim H k  log k 
k 
n
1 1

lim    log(k (k  1)) 
n 
2

k 1  k
n
1


 log 4n 
2 lim  
n 
 k 1 2k  1


1 
1 
lim  H n  log n 2  n   
2 
3 

n 
Euler
Cesaró
=
=
=
=
=
=
2

1   2
1
1 

lim H n  log  n  n     
n  
4  
2  45  


 1 
lim  n    
n 
 n 


1
 1 
1     log1   
 k 
k 2  k
=
Euler
log k
log 2
1 

(1) k

k
2
log 2 k  2

 (2k  1)  1
3
1  log   k
2 k 1 4 (2k  1)
 
1
1   j
k  2 j  2 jk
( 1) k  ( k )


k
k 2

 (k )  1
1 
k
k 2

=
Demys

1

1
  k  log1  k  
k 1

k 1
1
1 
 
= 
 ( k )  1    log 1   




k
k
k  1
k 2
k 2 
log 2   (2k  1)  1

= 1
2
2k  1
k 1

 (2k  1)  1
3
= 1  log   k
2 k 1 4 (2k  1)

 (2k  1)
= 1 
k 1 ( k  1)( 2k  1)

 (2k  1)  1
= 2  2 log 2  
k 1 ( k  1)( 2 k  1)

k  (k )  1
= 1  log 2   ( 1)
k
k 2

3
 (k )  1
 log 2   (1) k (k  1)
=
2
k
k 2

5
1
 (k )  1
 log 2   (1) k (k  2)
=
4
2 k 3
k

k  (k )  1
= log 8  3  2  ( 1)
k 1
k 2

4
k  (k )
= log  2  ( 1)
2k k

k 2
Euler


 
Euler (1769)
Euler
Euler-Stieltjes
Glaisher
Glaisher
Flajolet-Vardi
=
=
=
=
=

16
 (k )  1
1  log
 2  (1) k
9
2k k
k 2

3
1
1
 log 2     (k )  1  k 
2
2 
k 2 k 

11
1
1
1
 log 3     (k )  1  k  k 
6
2
3 
k 2 k 

 (k , s)
H s  log s  
k
k 2

1
(k )  1
 
,  (k )  log p if k is a power of a prime p
k
2 k 2

1    log n k 

  1


n  1 n!   k  2 k ( k  1) 

=
 2 k 1 1 (1) j 

k 



k
j
k 1  j  2


log k
(1) k  2 

k
k 1
Shamos

=
=
Vacca, Franklin
Vacca

log x
dx
x
0 e
=

=
1 

1
x  x 
dx
x2
1
=
1
  log log dx
x
0
=
  log x sin x dx
Andrews, p. 87
2
GR 4.381.3
0
 /2
=
 1  sec
2
x cos(tan( x ) 
0
dx
tan x
GR 3.716.12

=
 (1)    e  x log x dx
0
1
=
 H ( x) dx
0


 
 xe
x e x
=

=
1  e  x  e 1 / x
dx
0
x
dx
Prud. 2.3.18.6

1

 
Barnes

 
=
4
 2 log 2 


e
x2
log x dx
0

=
1
x dx
 2
2
2
(1  x )(e2 x  1)
0
1
=
 (1  e
x
dx
x
 e  1/ x )
0
1
=

0

=

1
x
0
=
1
  e  x dx
x
 1
  1  x  e
x
0

=
 1
GR 3.435.3
 dx
x
  1  x  cos x 


GR 3.783.2
 dx
 cos x 
 x
1
  1  x
2
0
1  cos x
0 x dx 
x
=
GR 3.427.2
 dx

 x
0
=
Andrews, p. 88
1

1
 
 dx
log(
1
)

x
x


  1  e

Andrews, p. 87


x
GR 3.783.2
cos x
dx  log x,
x

=
1  e x
0 x dx 
=
ab
a b
=
1   2k 
1 
  x  dx

0 1  x  k 1
=
1
x dx
 2  2x
2
 1)( x 2  1)
0 (e
x

e x
x x dx  log x,
e x  e x
0 x dx ,
a
x0
x0
b
a  0, b  0, a  b
1
Catalan


=
H 2n
 log n  2 
0 (e
2x
x dx
 1)( x 2  n 2 )
Hermite
Hermite

1 .57721566490153286061... 
 1 
 si ( x) log x dx 
GR 6.264.1
0

=
 Ei (  x) log x dx 
0
.577350269189625764509... 
3

 tan 
3
6
GR 6.234
=
4
3


 
=

1
 (2k  1)
k 1
 2k 
1
  k  6 (k  1)(k  2)
k
k 0
.577779867999970432229... 
.577863674895460858955... 

 
=
 (2)

 ( 2 )   ( 3)


 2
  1  
k
2e
k 1 

cos x
0 1  x 2 dx

=
GR 1.421
1/ 9
2
cos x
 (1  x )
2 2
( 1) k  1 k

AS 4.3.146
dx
0

=
xsin x
 1 x
2
dx 
0
 /2

=
 cos(tan x ) cos
2
x dx
GR 3.716.5
0

=
 cos(tan x)
0
1
=
sin x
dx
x

GR 3.881.4
1  dx
  cos x  cos x  1  x
Prud. 2.5.29.11
2
0

=
x 3 sin x
 (x 2  1)2 dx

27 log 3  2
 3



6
6
2
 2  3  log 2 x
1 .57815429172342929876... 
 
2
3e
0 ( x  e)
.5781221858069403989...


.578169737606879079622... 
 (2k )
 (2k )
k 1
.578477579667136838318... 

 
=
2 .578733971888556748934... 
4 .57891954339760069657... 
2

Marsden p. 259

 3k
k 1
3
1

 k2
1 
 1
Li2  2 

4 k 1  k 
 (sin  2  sinh  2 )
1


2 2 (cosh  2  cos 2 ) 2

1

1
k 1

1
1


k 1
  ( 1)  (4 k )  1  Im 2 2

2 k 1
 k  1 k ( k  i) 

 3 (k )
k 2
k!

4 log 
k
4
.578947368421052631
=
.57911600484428752980... 

11
19
2

4
 7 ( 3) 

k2


1 4
k 1 ( k  2 )
2
24
2
4
1 .57913670417429737901... 

25

1
1
1
1


 
=  
2 
2 
2 
2
(5k  2)
(5k  3)
(5k  4) 
k  1  (5k  1)
6 .57925121201010099506...  log 6!

3
 3
1  2 log 2      ( k )  1 

 
2
k 2 2

1


k  2 k ( k ! 1)


1
 1 ( k )



k
3k
k 1 (3  1)k
k 1
 x
2 2
e  e x  2 2
x dx
2  
3
( e x  1) 2
0
2
3 .579441541679835928252... 
.579480494370491801315... 
.5795933814359071842...

4 .57973626739290574589... 

2 2

3
6 .57973626739290574589... 
 log
2
x
0

 4k
k 2
2
9
 6k
GR 3.424.5
dx
(1  x ) 2
GR 4.261.5
 2
 Li2   
 3
4 .579827970886095075273...  5G
.57975438341923839722... 
2 .58013558949369127019... 
.580374238609371307856... 
5 .580400372508844443633... 
.58043769548440223673... 

 

 ( ( ( (2))))


1
k


(1) k !  (5k  1)  1


5
k

1
2
k 1
k 1
k

e

k 
k 1 k


 1 
arctan(tanh )   ( 1) k 1 arctan

 2 k  1
4
k 0


.580551791621945988898... 
 ( 1)
k 1
2e 
(k )
3k  1
k 1
6 .580885991017920970852... 
[Ramanujan] Berndt Ch. 2

2
1/ k !
k 0
.58089172936454176272... 

3
 1
 (3)  Li3    
 3
4
1
log 2 x
0 ( x  1)( x  3) dx 
.58105827123019180794... 

2

cos
 3




1
 1  2k (k  1) 


k
1
3  log 2    2  (k )  1
2
k 2 k  1
2
k 2

.58106146679532725822... 

.581322523042366933938... 
2
dx
 3
13 / 6
3
0 x 3


  2  2i 3 
1/ 6





 cot
 i coth   
.581343706060247836785... 
cot   1




2




2


1
k
 
=  6
   (1) k 1  ( 6k  2 )  1
2 k 1
k 1 k  1
1/ 4

29 e
k3
.581824016936632859971... 
 
k
64
k 1 k ! 4
i
6



1
k3
.581832832827806506555...   7
   ( 1) k 1  ( 7 k  3)  1
2 k 1
k 1 k  1


1
1
1
.5819767068693264244... 

 k   k
1 / 2k
e  1 k 1 e
k 1 2 1  1 / e

 



[Ramanujan] Berndt Ch. 31
(1) Bk
k!
k 1

1 

1  2 k 

e 
k 0 

log x
1 ( x  e  1)2 dx 

=
=
=
2 .581931792882360146140... 

k

Prud. 6.2.3.1
 5  2    log1  x 3 2  dx 


1 
  2  1

 log1 
 dx


2
0

.581954706951074968255... 

5
0

1 .581976706869326424385... 
e

e 1
3 .582168726084073272274... 
 2  2 e cosh
1

k 
k 0 e

5x2  1 
(1) k Bk


k!
k 0


5

2

Lk
 k!
k 1

.5822405264650125059...

 2 log 2 2
1  1

 Li2     k 2
12
2
 2  k 1 2 k

J362


 
=
 ( 1)
k 1


 
Hk

k
1
k 1
H ( 2) k
= 
k2
k 1

log(1  x )
0 x(1  x) dx 
1
=


 
=
x
J116
MIS
2
log x
dx
2
x
x
1
GR 4.291.12
dx
 log x  1  x
2
log(1  x )
dx
x
0
1/ 2

=

=
x  dx

  log 1  

2 x
0
GR 4.291.4
=
 1  x  dx
  log

 2  1 x
0
GR 4.291.5
GR 4.291.3
1
1
4 .58257569495584000659... 

1 .58283662444715458907... 
.583121808061637560277... 
21 
3
sin  2 csc 3 
2
2G

 1  k
k 1
2
1


 4k  1 

.58326324064259403627... 
sin 1 log 2
.583333333333333333333 =
7

12

.58349565395417428579... 

 ( 1)
k 1
 1 ( k )

2k  1

3e  1
2k k 2
 

2
k  0 ( k  1)!
k 1
2
10 .583584148395975340846... 

.5838531634528576130...

2 cos2 1  2   ( 1) k 1
k 1

1 .584893192461113485202... 
101/ 5 
1 .58496250072115618145... 
log 2 3
log(1  x ) log x
dx
x
0
1
.5852181544310789058...

16    8 log 2 
2

1 .58556188208527645146... 
1
 2
k 2
(k )
22k

( 2 k )!

GR 1.412.2
(1) k
.585698861949791886453...   3
k 0 k  1

.585786437626904951198... 
.58617565462430550835... 
2 2 
1 2
 /4


0
 7 4  4 
      ,1 
 3 3  3 
dx
sin x  1
1
e
 x 3/ 4
dx
0
 0 (k )

.586186075393236288392... 
2
 (2k )!
k 1
9
8
2
4 .586419093909772141909...    1
8 .586241294510575299961... 
.586781998766982115844... 
.586863910482303846601... 
 
1
2

arccsch 2 
3 3 3
HypPFQ{},{2,3},1 

 (1)k 1
k 1

2k
 2k 
 
 k
1
 k!(k  1)!(k  2)!
k 0
1 .58686891204429363056... 
2 .586899392477790854402... 
.58698262938250601998... 
7 .587073064508709895971... 
G

 5 
512
 

63
 1 / 2

tanh

2
8
 3
1  2  / 2
cos
e  e

3
3
2

log 3 1 (1)  2 

   

2
9
 3 6 3
8
7 .58751808926722988286...     
7

k 0
k (k )

3k
k 2


sinh 1  k
 1  dx

  cosh 2  3
2
x x
k 1 ( 2k )!
1


dx
3 2
2
0

3
.587785252292473129169...  sin
 cos 
5
10
.587719754150346292672... 
 (3k )!
1
.587436851334876027264... 
.58760059682190072844... 
 3k
log( k  1)
log( 2  x )
 
dx
k
2 k
log x
k 1
0

.587413281200730307975... 

5/ 6

x
6
GR 422.3

6k  1
 3k (3k  1)
k 1
2

.58778666490211900819... 
.5882352941176470
=
log
9

5

Lk
k
k
3
k 1
10
17
 2 k  1 k 2
9 3  3  

k  6k
k 0

12 .588457268119895641747... 
1 .588467085518637731166... 

 
=


i  (1) (1  i )   (1) (1  i )  i  2,2  i    2,2  i  

 1
1 

i  

2
(k  i ) 2 
k 1  ( k  i )

.58863680070958604544... 
 ( 1)
k 1
k 1
Fk  (2 k )  1

3  3

(1) k  1

csch
  2
2
2
3
k 1 k  1 / 3

1
.588718946888426853122...  2 I 2 2  
k
k  0 k!( k  2)!2
.588645903311846921111... 
 
3 

log 2 x
0 ex 2  1 dx
8 e

3
k ( k  2)
.589048622548086232212... 
 
3
1
16
k 1 ( k  2 )( k  2 )
2 .58896621435810891117... 



 
=
sin 5 x
0 x dx 

=
sin 6 x
0 x 2 dx 
3
cos x sin x
dx 
0
x

=

cos 4 x sin 2 x
dx 
0
x2

J1061

 (x
0
dx
 1)3
4
cos x sin x
dx 
0
x

cos 4 x sin 2 x
dx 
0
x2
(1) k 
.5891600374288587219...  8  2   ( 2)  4 log 2   3
2
k 1 k  k / 2
7
6 .58921553992655506549...    
 6

  ( 2 )k


1
5 .589246332394602395131...  e1 
erf

 k  0 k!! 
2


 /4
5
.589255650988789603667... 
  cos3 x dx
6 2
0


log 2 k
.589296304394759574549...  

2k
k 2
GR 3.827.12
2

cos3 x sin 2 x
dx
0
x2
1 .58957531255118599032... 
2 .589805315924375704933... 
.589892743258550871445... 

 1 
log      
 4


 e
e

   log x log 1  2  dx

x 
0
2

 4  21 / 335 / 6 i  61 / 3 
1 
1/ 3






(
1
3
)

2

(
2
6
)
i




65 / 3 
2



1/ 3 5 / 6
1/ 3
(1  i 3 )  4  2 3 i  6 


 
65 / 3
2



=
.59000116354468792989... 
1
6
k 2

S 2 ( 2k , k )

2
k 1 ( 2k )!k
k
3
( 1) k  ( k )  1
 

2k  3
k 2

.590159304371654323...
(1) k Bk
1
 

6
k  0 ( k  2)!

.59023206296500030164... 
3
.590320061795601048860...  24
 18 
5
9
3 .59048052361289410146... 
2

 . 59048927088638507516... 

 

.590574872831670752346... 
1 .590636854637329063382... 
4 .59084371199880305321... 
4 .59117429878527614807... 

 2


1
1

3 arctan
k
k
 
k 2
(1) k Bk

k  0 ( k  2)!

 2 k  2 ( 1) k




k  6k
k 0 

 
3 2
arctan 2 
4
2
2
1
arctan x 2  1
0 ( x 2  1) x 2  1 dx 




Borwein-Devlin, p. 53
G6
I1 (2) 

1


k  0 k!( k  1)!

k

2
k 1 ( k!)

 2k  k
  k  (2k )!
k 1


LY 6.113
1
 
5

2
2
 
2  arcsinh 1 
4  x 2 dx
0
.591330695745078587171... 
 log
1 2

2
1
 log(1  x
0
2
)
dx
1  x2
GR 4.295.38
 /2
=
 (1  sin
2
x ) dx
GR 4.226.2
0
13 .59140914229522617680... 

k3

k 1 k!
5e 
J161
1 

 1
  2 log 2    li  sin( 2 log x ) dx

 x
5 2
0
1
.591418137582957447613... 
14 .591429305318978476834... 
1 .591549430918953357689... 
5 .591582441177750776537... 


9 .5916630466254390832...
 2 3 csc  3 
6
5
  
92
6 log 3 
e
dx
x
2
0
1 .59178884564103357816... 
10 .5919532755215206278...

 e x
2

 1 
2e erf 
 
 2
2 sinh
2

2
erndt Ch. 28, Eq. 21.4

 (k 
k 0
 
1
2
=


4
1 
 

2 
(2k  1) 
k 1 

)!2 k

 2k
 (2k )! 
k 1
e e
 cosh   cos i 
2


1
 cosh   1 

11 .5919532755215206278...
k 2  2k  1

2
k 1 k  2 k  2



.59172614725621914047... 
1


2
k  2 1  3k
5

6 .59167373200865814837... 

GR 6.213.2


 2k
k 0
(2k )!

k2  4

2
k 1 k  1
AS 4.5.63

J1079
7
182 (3)  4 3 3  432    ( 2)  
6
3
( 2)  1 
433 .59214263031556513132...  182 (3)  4 3     
6


2
 1/ 4 1
k!
.59229653646932657566... 
e erf  
  e  x sinh x dx 
2
2 k 1 (2k )! 0
1 .59214263031556513132... 

.592394075923426897354... 
k 2
.59283762069794257656... 

  1 
 (k  2) 

 (k ) 
2 sin 1
sin 1


5  4 cos1
2(1  cos1  1 / 4)

sin k

k
k 1 2

GR 1.447.1
1 .5930240705604336815...

=
.593100317882891074703... 
log 2
2
2 2
log 2 log(3  2 2 )
log




2
2
2
2 2
2

H 2 k 1

2k
k 1


1
3

  2
2 2 3
k 1 k  1 / 3
.59313427658358143675... 
 (2) 
.593350269487182069140... 

2


.593362978994233334116... 
3 .593427941774942960255... 
.593994150290161924318... 
.594197552889060608131... 
cos
H
2
k 1
7 (3)

8
k
( 3)
k
 (3) 
 5
log 2 x
1 ( x  1)( x  1)2 dx 


k 2

1

 1  k (k  1) 

2


k
2
 2 log 2
log3 2


3
3

3
Hk


k
k 1 2
1  3e 2
i
1
 i
2 sin 1
e  ei
2
( 1) k 1

4
2
12
k 1 k  k

(1) k
.594534891891835618022...  1  log 2  log 3  1   k
k 1 2 k
.59449608840624638119... 
 csch 
.594696079786514810321... 
 1

1

2

3i 3 5i 3 3i 3 5i 3


 
 


6
2
6
2






 
=
  (3k  1)   (3k  1)
k 1
.594885082800512587395... 
4 e 6 

1
 (k  2)!2
k 0
2 .594885082800512587395... 
4 e4 


k 1
pf (k  1)
2k
 0 (k )
k 1 k !1
 5
 
 4

1 .594963111240958733397... 
4 .59511182584294338069... 


.59530715857726908925... 
  (k ) log  (2k )
k 1
k
4
2
1 .59576912160573071176... 
2 k 1
3
1 
k 1 3
(
1
)



4 k 1
(2k  1)!

H
.595886193680811429201...  3  2 (3)   2 k

k  2 k ( k  1)
1
   (1     H
.596111293068951500191... 
.5958232365909555745...

3 .59627499972915819809... 
.59634736232319407434... 

 
=

 
=
sin 3 1 

 log 

 ( k  1)
k!
k 0


2
dx
x dx
0 e x ( x  1)  0 e x ( x  1) 


 e Ei ( 1) 

dx

2
x (1  log x )

1
GR 1.412.3
, the Gompertz constant

e
x
log(1  x ) dx
0

x 4 dx
4 .59634736232319407434...  4  e Ei ( 1)   x

e ( x  1)
0
( 1)k 1 k

k
k 1 2  1

.5963475204797203166...

log 2 1


2
4

1 

1 .59688720677545620285...    1  k 2 
2 
k 1 
.596573590279972654709... 

x log x
 (1  x)
3
dx 
1
(1) k 1
.596965555578483224579...  

k 1
k 1 k


.59713559316352851048... 
80

243 3
e 5

4

.597264024732662556808... 
2k

k 1 ( k  2)!
e2  1

4

1 .597264024732662556808... 
2

1 .597948797240904373890... 
0
dx
 1) 4
3
2k

k  0 k!( k  3)
 (2k )   (2k  2)


k!
k 1

.597446822713573534080... 
 (x

k 2


 k2 1 1/ k2

 2 e
 1 

k 2  k



1 1
k  (k )  1    Li1 / 2    
k k
k 2 




1
cos 4k
 4i
4i
.598064144464784678174...   log ( 2 sin 2)  log 2  e  e
 
2
k
k 1
1 
erf 2 
2 2
.59814400666130410147... 
54 .598150033144239078110... 
1 .598313166927325339078... 
8 .59866457725355536964... 
.599002264993457570659... 
4k

k  0 k!
4
2
.599314365613468571533... 

2
2

127

128

k
2
k 1
k

k 2


3
4
  k  1, 
1
 (3)
3
 log (2 k ) 
2  e    Ei(1) 


1
.599971479517856975792... 

k 1
arccsch

2
k2
 (k  1)! k   (k  1)!(k  1) 
17 4
17 (4)



360
4

sin 2 ( k  1)

k  1 k ( k  1)

 log (3)  1
k 1
1
 
= 

k  2 ( k  1)!  2 k !


Hk
1
= 
 
k  1 k !( k  2)
k  1 ( k  1)! k
k

2
1 .599771198411726589334...  
k
k  1 ( k  1)
 
8k 6  5k 4  4k 2  1

k 2 (k 2  1) 4
k 2

k

4 .599873743272337314...


 k 3  (2k )  1 
1
k 1
.5996203229953586595...

 k! F

.59939538416978169923... 
48



  1 cos1 sin 1  sin 2 1  log( 2 sin 1) sin 2 1 
2 
k 1
1 .599205922440719577791... 

4G 

1 .59907902990129290008... 

e4 
120
(1) k 2k

k  0 k!( 2k  1)


2
 Hk 
 


k 1  k 
Borwein & Borwein, Proc. AMS 123, 4 (1995) 1191-1198



.600000000000000000000 =
3

5

8

5

1 .600000000000000000000 =
.600053813264123766513... 
F2 k 1
k
k 1 4
Lk Lk
k
k 1 4

3 5 2 log 2 2
 1  i
 1  i


 2 Li2 
  2 Li2 

 2 
 2 
2
24
2

=
log(1  x 2 )
0 x 2 (1  x )2 dx
(1) k  1
.600281176062325408286...   3  6 log 2  9  

k  1 k ( k  1 / 3)

.6004824307923120424...

2 .600894716163993447115... 

 (2k )   (2k  2)
k 1

k

e
1/ k !
k 1
.601028451579797142699... 
 (3)
2


 
=

 1 

H k H k 1
 k (k  1)(k  2) 
k 1
log 2 xdx
log 2 (1  x)dx
x dx




0 e x sinh x 0 sinh(log x) 1 sinh(log(1  x)) 
1
2

.60178128264873772913... 
 (1)k 1
0
 (k  1)  1
2
k
1
k
.601782458374805167143...  
log
k 1
k 2 k  1
k 1


.601907230197234574738... 
k2 1
k2 1
 
log
k
k
k 2

K1 (1)   e 
1 x 2

1  1
   Li2   
 k
k 2 k
dx
0
.602056903159594285399... 
 ( 3) 
.602059991327962390428... 
log10 4
74 
sinh  2
1
2 .60243495809669391311... 

csch  sinh  2 
2 sinh 
2


1 
k2  2

 
=  2
  1  2

k  1
k 1 k  1
k 1 


1
1
.602482584806786886836... 
coth  5 
  2
10
2 5
k 1 k  5
8 .6023252670426267717...

3
5

2 4 5 2
175 .602516306812280539986...  24 5)  50 ( 3) 

1 
3
2

16k 4  k 3  11k 2  5k  1
= 
k ( k  1)5
k 2

2 sinh
1 .60343114675605016067... 
 2  2 2 arctan
 
=
Berndt 7.2.5

2
 log1  ( x  1)

6 2

k 2
2
2
0
2 .603451532529716432282... 
4

4 .602597804614589746926... 


 k  (k )  1
1
 log 3
2

 dx

sin  2 



2
 1  k (k  4) 

7 (3) 1
k3
   2
2 .603599580529289999450... 
3
4
2
k 1 ( k  1 / 4)

1

2 .603611904599514233302...    1  k 
k 
k 1 

k 1


.603782862791487988416... 
1 .603912052520052227157... 
.60393978817538095427... 
1 .604303978912866704254... 
log k

k 2 k !

1
2
1
1
  /2 
2 2e

.604599788078072616865... 
=
=
=

2
k4  k2 1
  4
k  k2
k 1

e
x
cos x dx 
0

  log1  x 2 1  dx


3
0
 1 (k )
2
k 1
3
 /2
2 1 / 3
3 1 
3

3 .60444734197194674489... 

1  i  1  i  cosh 2
k
1

1 
 1
 


3 3 k 1  3k  2 3k  1 


( 2 k )!!
 
k
3 3
k 1 ( 2 k  1)!! 4 k

1

k  0 3k  1)( 3k  2 )

1

k 1  2 k 
 k
k

dx

= 
2
2x  2x  2
0

dx
0 4 x 2  2 x  1 
CFG F17

x
0
4
x dx
 x2  1

=
=
0



x
dx
 3x  3

e

x dx

3
8
x
0
4
0
dx
 e x  1
x
0
10 .604602902745250228417... 
0
2
dx
 x  x3  x2  x  1
5

=
x
dx
x dx
x 3 dx
1 x3  x 2  x  0 1  x6  0 1  x6 

=

dx

2
x  2x  4
log 8!
1  2   12     12 , 12     12    (1)k
k 1

.604898643421630370247... 
 


 
k 1
10
5 .604991216397928699311... 
.605008895087302501620... 
7 2  3
9 log 2 3 1 (1)  2 

log 3 
   
 3
48
8
16
4
.605065933151773563528... 
9 2


4
2
.605133652503344581744... 
1
 erfi(1)  Ei(1)    x 2 dx
2e
0 e ( x  1)
1  x  x2
0 x  1 log x dx 
1

.605139614580041017937... 
1 .605412976802694848577... 
.605509265700126543353... 
k2

k
2
k  2 2 ( k  1)

(1) k 2 2 k  1
si ( 2 )  
k  0 ( 2 k  1)!( 2 k  1)
3 ( 2 )  4 ( 4 )

1

.6055217888826004477...   2
k  2 k log k
3 .605551275463989293119... 
.605591341412105862802... 
.6056998670788134288...

2 .60584009468462928581... 

9 3 log 2


8
4
AS 5.2.14

  ( x)  1dx
2
13
5 3

256
sin( 3k / 4 )
k3
k 1


GR 1.443.5
 2


sin 
 1  cos

2
 2

 1  (2k  1)

1
2
6
1
 2 log 2 2 


k 1


 log 1  x dx
2
0
2
2



.606125795769702052589... 
3 .606170709478782856199... 


1
1   
4 
e
3 (3) 

e
x2
cos2 x dx
0
HkHk
 k (k  1)
k 1
1 1
=
log( x 2 y 2 )
dx dy
1  xy
0

0
.606501028604649267054... 
.606530659712633423604... 
15 .606641563575290687132... 
 (1)
k 1
 (2k )  1


k!
1
 ii / 
e
 1  e
 1/ k 2
k 2
(1) k

k 
k  0 k !2


(1) k

k  0 ( 2 k )!!

 5 

1
( 2 k )
.60669515241529176378...   k 1

  k k
 
2k
1
k 1 2
k 1 2 ( 2  1)
k 2



1
 (k )
2k  1
  0k
  k2 k
1 .60669515241529176378...   k

k  1 2 ( 2  1)
k 1 2  1
k 1 2


1




(2 k )


2k
k 2

 
=
1
 1


jk
j 1 k 1 2

 

Shown irrational by Erdös in J. Indian Math, Soc. (N.S.) 12 (1948) 63-66
.606789763508705511269... 



1 .606984244848816274257... 

1
2

  log 2   
0
xe x
ex  1
dx
GR 3.452.3
1
 k ! (k )
k 1

erf (1)
( 1) k
 

1
e
k  0 ( k  2 )!
(5)  1
3 .6071833342396650534...  3(3) 


1024 k 1 a( k )5
  where a ( k ) is the nearest integer to 3 k .
.60715770584139372912... 


AMM 101, 6, p. 579

k4

k  0 k !( k  4 )

 2 k  2
1
1 .607695154586736238835...  12  6 3   
 k
k  6 ( k  1)
k 0
5 .607330577557532492158... 
.607927101854026628663... 
1536  563e 
6

2

1

 (2)


k 1
 (k )
k
2


1
2

 1  p
p prime
.607964336255266831093... 
.60798640550036075618... 
4 12

6081075
 
=
x
2
2


 
=
.608197662162246572967... 
3
3
log 
2
2


 
=
.608531731029057175381... 
k 1
2 k 1
k

(2k  1)
Hk
k
k 1
1
 ( k  1)!2  (2 k )
Titchmarsh 14.32.1
k

1 
p2
 2
 2  log p 
2
p  1
p prime
p prime  ( p  p  1)( p  1)

  (2)
( p  2) log p
 2
 

2
 ( 2)
p prime  ( p  p  1)( p  1) 

log p

p  p 1

2
2 8

31185

p prime
1
1 p
2
p
4
 p 6  p8


sinh 
e  e  
 (2 k )  1

 log
  ( 1) k 1
4
2
k
k 1

1
1

 
= log
  log 1  2 

 (2  i ) (2  i )
k 
k 2

 2k   ( 2k )  1
1 .6089918989086720365...    
k!
k 1  k 

 (k )  1
.609023896230194382244...  
Hk
k 2
.608699218043767368353... 


2
3
k 1

p
1

 p  p  8  p 10  p 12
6
(sin x  x cos x)3
dx
0
x4
 ( 2 )   (5)
.608381717863324722684... 
1 p
4
dx

 x2  2
.608006311704856510141... 
.608199697591539684584... 
p prime
2
log 3 1
1 1 3 1
  2 F1  ,2, ,  
4
3
2  2 2 4



log

.609245969064096382192... 
4 .609265757423133079297... 
Hk 3

k 
k 1 4
  2 csc  2 

k 2
1 .609437912434100374600... 
.609475708248730032535... 
 4
log5  Li1  
 5
2 2 1
 
3
3
 /4

0
1
 1  2k
sin x
dx 
cos4 x
2
K Ex. 124f

(1) k

.6100364084437080...

 k (k  1) log k
k 2
7 .610125138662288363419... 
.610229474375159889916... 
1
cosh e   ee  e e  
2
2
e2 k

k  0 ( 2 k )!

4
x3
dx

 9 (3)   x
(e  1) 3
2
15
0

.610350763456124924754... 


1
 log 1  k ! 
k 2
2G
3
2
.61094390106934977277...  8  e
11
H
.611111111111111111111 =
 3 
18
3
.6106437294514793434...

.611643938224066294941... 

 
=

1


2
k  1 k  3k
k
k 4
2
1

 3k
  1  i 3
  1  i 3 
1 
1
  (1  i 3 ) 
  
   (1  i 3 ) 
2 3 6
2
2







k


3
k 2 k  1

 (1)  (3k  1)  1
k 1
k 1
2
1953125
5 5 
 
  log
 5 log
5  1048576
5  5 

14 4
log 3 x dx
5 .6120463970207165486...  
  3

243
x 1
0


1 .612273774471087972438... 
2 .61237534868548834335... 

 
=
.61251880056266508858... 
sinh 


1 
2

 1   k
k 1

1
 1
 3


3/ 2
 2
 2
k 1 k
1  p 3 / 2
  (6)  1  p  3 / 2  p  3  p  9 / 4

3
p prime 1  p
p prime

14 
2 2  4

 3 ( 3)   (5) 
3
45

  1  
k 2
42 .612923374243529926954... 

k 1
     4    

1 .612910874202944353144... 
 0
   
 2i
Fk Fk 3
k
(k  1)
4
.612018841644809204554... 
3

1
 (k  1)
k 2
1 

(k ) 
sinh 2
1
 cosh  sinh  
2



4
2 

 1  k
k 1
2

1 1
  5
k k 
.613095585441758535407... 
.61314719276545841313... 
3 .613387333659139458760... 
2 .613514090166737576533... 
.613705638880109381166... 

 
=
=
=
1
  1
log tan   
 4 2
2
log 6 3 

 (k  2)
k 0
k!

 (1)
k 1
k 1
sin( 2 k  1)
2k  1
GR 1.442.3

e1/ k
  2
k 1 k

 ( 3)
 3
 1
2  2 log 2        hg   
 2
 2

1

2
k 1 2 k  k

k

k
k  0 2 ( k  1)


2k  1
k  1  ( k  1)
(

1
)

 ( k  1)  1


k
2k
k 1
k 1 2
1
=

log x
 1 x
2
GR 0.234.8
dx
0
1
=
  log(1  x 2 ) dx
=
4 log(1  x )
0 (x  1)2 dx 
0
1
1
log(1  x )
 (x / 2  1 / 2)
2
dx
GR 4.291.14
0
 /2
=

 log(sin 2x ) sin x dx
GR 4.384.10
0

k3
4 .61370563888010938117...  6  2 log 2   k
k  1 2 ( k  1)

k5
128 .61370563888010938117...  130  2 log 2   k
k  1 2 ( k  1)
.613835953026141526250... 
.613955913179956659743... 
.61397358864975783997... 
2 .61406381540519800021... 

11
2
tanh
7 / 10

 11
2


5 5 5
1
log 2  2
2

4 log 2 

x
0
dx
2
5

4
k 1
1
 
3
( 1) k 1 / k


k
k 1
2
1
k3

.614787613714727541536... 


1
 1  k (3k  1) 
k 1
1
2 
1
.61495209469651098084....  erfi


2 k 1
2
 k  0 k ! 2 ( 2 k  1)
8
.615384615384615384
=
13
 /4
3
cos x dx
1
.61547970867038734107...  arctan
 arcsin
 
2
5
2
0 1  sin x
( 1) k 1 2 2 k 1 ( 2 2 k  1) B2 k
 log(cos1)  
AS 4.3.72
( 2 k )! k
k 1
58 
1
1
1
 1 

 sin  sin  2  cos   cos sinh  2 sin 
1
  2 
2
2
2 1

 cot 
1
1
2
2
2


cos  2 cos   cosh  2 sin 
2
2


i
1
  e  i / 2 cot e  i / 2  ei cot ei / 2  2i cot  
2
4

.615626470386014262147... 
7 .6157731058639082857...

.61594497904805006741... 

 
=
 

 




  (2k )  1 sin k 
Adamchik-Srivastava 2.28
k 1

arctan k
k!
k 1
 8 ( 3)
1 .61611037054142350347... 
9 .6164552252767542832...
54 .616465672032973258404... 
1 .6168066722416746633...


2 cosh 4  e4  e4

 2 ( 1)
2 log 2 
k 1
/k

k 1
.616850275068084913677... 

(1) H k


16 k  0 k  1
2

k

2
1/ 2 k k
k 1
k (2k )


4k
k 1


4k 2


2
k 1 ( 4k  1)


 
=
5 .61690130799559924468... 
1 .61705071484731409581... 
.617370845099252888895... 
x arctan x
dx
1  x4
0

GR 4.531.8
( 1) k

1
1
k  0 ( k  2 )( k  3 )


3 1
1
coth 
   2
2 3
6
k 0 k  3 / 2
2  3  6 log 2  3 
sin 2 
cos 2 1

2
2

1 .617535655787233699013... 
5 log
 2 k  1
1
 k
k  5 ( k  1)
k 0

 
1   1  5 
1  
  Li2  
Li2 
 
5   4 
 1  5  
.617617831513553390411... 
4 .617725319212262884852... 
10

5 5

8

Fk
k 2
k
2
k 1



.61779030404890075888... 
 k  1  (k  1)  1
k
k 2

1 .61779030404890075888... 
k2
 (k  1)  1 

k 2 k  1

 3k  2
  k (k  1)
k 2


2
1
k 

log
k
k  1 

3 .6178575491454110364...

2

2
 2 log 2
2

7 (3)

4

  2k 
4k
k 0
  (k  1) 2
k 

15
3 log 3
7
 

  
.617966219979193677004... 
4
2
2 3
 3


5
dx
dx
2 .61799387799149436539... 

 x
12 / 5
x
6
 3
0 1 x
0 e e
.618033988749894848204... 
 1 
1


5 1
2

5 1
 2 cos , the Golden ratio
2
5
2
2 .618033988749894848204...    1  

1
.618107596152659484271...   k
k  1 2 kH k

 (k  2)
1 .618157392436715516456...  4 log 2  2  
2k
k 0
2 4
1
.61847041926350753801... 
 
2
315
 p 4
p prime 1  p
1 .618033988749894848204... 
.61851608883629551823... 
1 .618793194732580219157... 

.61897071820759046667... 

3 9 log 2


8
4
1
2
 3
cosh

3
2


(1) k 3k  (k )

4k
k 2



k 1


1
 1  (k  1)(k  2)  

1
10   5  5 H (1
10

5) / 2


 5  5 H (1
5) / 2



 
=
 F  (k )  1
k 2
k 2
.619288125423016503994... 
16 10 2


3
3
  k  k
k 1
.619718678202224860368... 
k
k 0
1
(k  2)

6
k
cot

6

 3k

k 1

2
1

1/ 2
(1) k 1

2
2
6
2
k 1 k ( k  1 / 2)

(1  2 k ) (1  k )
e 1
.62011450695827752463...  log

 
k!
2
k 1

(1) k (2k  1) Bk
 
= 
[Ramanujan] Berndt Ch. 5
k !k
k 1
.620088807189567689156... 

1

  k  8
 2k  1

4 .619704025521212812366... 
 2k 

2
1
=
  2 csch

2 
dx
 1 e
x
0

4 e1/ 3
k2
 
k 
9
k 1 k ! 3

 0 (k )
4 .620324229254256923750...  
k  1 ( k  1)!

k 1  (2 k )  1
.62047113489033260164...   ( 1)

2k  1
k 1
.620272188927150901613... 
.62073133866424427034... 
8 .62074214840238569958... 

 

24
 29

cos
7
2

k 1

1
1
 k arctan k
k 2
1

 1 k (k  5) 


 4  i 21/ 335/ 6  61/ 3 
1 
1/ 3

  2 2 / 3  2  61/ 3  
1
(

)


2/3
3 
2



(1) 2 / 3  4  i 21/ 335/ 6  61/ 3 

 1/ 3  
3
2




=


 6  (3k  1)  1
k
k 1


k
k 2
6k
6
3
 1 1 
, ,0 

 2 2 
/2
9 .62095476193070331095...  2 e
1 .62087390360396657265... 
.621138631605517464637... 
2
24


2
 log 2 
log 2 2
 1  i
 1  i
 Li2 
  Li2 

 2 
 2 
2




2
3 log 2 2
 
=

 log 2 

2 16
4
1
log(1  x 2 )
 
=  2
x ( x  1)
0
1
.621334934559611810707... 
 log5 e 
log 5
1
 5  2  13  4 5  50  12 5  2 65  20 5  
2 .621408383075861505699... 

4
 
5  5  5  5  5  5  5  ... [Ramanujan] Berndt Ch. 22
=
.621449624235813357639... 


2
cos1 ci(1) sin 1 ci(1) cos1  ci(1) sin 1 

=
sin x dx
0 1  x 


 
=
cos1
  2si(1)  
2

arctan x dx

ex
0

dx

2
 1
 e x
x
0
( 1) k

k
k  2 2  ( k )  1

.62182053074983197934... 
2 3
6
1
1
 2   , lemniscate constant
2 .62205755429211981046... 
2 2  4 
1
1  3e2  3e4  e6
3
1 .623067836619624382082...  sinh 1   sinh 3  3 sinh 1 
4
8e3

( 3k  3)(1  (1) k )
= 
8k !
k 1
2 k 1

1
3
3
=

4 k 1 ( 2 k  1)!
1
1
1
1
log 1  2 
arcsinh1 
arctanh
.623225240140230513394... 

2
2
2
2

 (1) k
(1) k 

 
= 1  

4 k  1
k 1  4k  1
.622008467928146215588... 



=
2
k 1
k
1
( 2 k  1)
CFG G1
J883

GR 1.513.1
(1) k ( 2 k )!!

k  0 ( 2 k  1)!!

(1) ( k 1) / 2 

2k  1
k 0

1
1
Lik  

k
 2
k 1 2
1
x dx

=
=
=
=
 (1  x
2
0

=
 2x
2
0
 /4
=
=

sin x
dx
2  cos2 x
0
=
x
2

.623263879436410617495... 
dx
2
2
 2k
k 0
.623658674932063698052... 
Adamchik-Srivastava 4.12
dx
 4x  1
dx
sin x  cos x

Prud. 5.1.4.4
) 1  x2

0
 /2
J129
1
3
3
3  3
  
4  2 

 e
0
dx
x2/3
1
1 3
2

pr ( k )
.623887006992315787...
  k
k 1 2  1

13078
k 6F
104 .624000000000000000000 =
  (1) k 1 k k
125
2
k 1
.623810716364871399208... 
2 log
.624063589774511570688... 
1

 3

coth

6 4 3
2

 4k
k 0
4
1
2
3
2
5 2

4 
3
2

1 
=   ( 4k  2)   ( 4k  1)  
   (4k  1)   (4k ) 
2 k 1
k 1

k2  k 1
=  4
3
2
k 2 k  k  k  k

2 4 5 2
3 .624270016496295506006...   24 (5)  50 (3) 

 1   (1) k k 4  (k )  1
3
2
k 2
.624270016496295506006... 
 24 (5)  50 (3) 
16k 4  k 3  11k 2  5k  1

k (k  1)5
k 2

=
1 .624838898635177482811... 
K 2 (1)  K 0 (1)  2 K1 (1) 

=
x
2
e
1 x 2
0
.624841238258061424793... 

 ( 3)  

dx
.625000000000000000000 =
.62519689192003630759... 
.6252442188407330907...

5
8
2  8 3
 

3 2
27

log
csc
3


3

1 

1 .62535489744912819254...   1  k 2 
2 k 
k 1 
 1
3 .625609908221908311931...    
 4
 /2

0

 ( 2k )
k 1
3k k

2 k  ( k )  1


k!
k 2
.626020165626073811544... 

2
sech

2


  e
 2k
k 2

 /2


AS 6.1.10

1 .625789575714297741048... 
sin2 x
dx
(2  sin x ) 2
e

e 1

2
 1 
k
cos x
 cosh x dx
GR 3.981.3
0

cos x
dx
 e x



 1
k 1  (2 k )  1
.626059428206128022755...   ( 1)
   Li2   
2
 k 
k
k 1
k 2
=
e
x

.62640943473787862544... 
.626503677833347983808... 
k

3
k 1 ( 2k  3)

6 sin 1
k sin k



2
(5  4 cos1)
2k
k 1
 2  16  21 (3)  16

log 2 x dx
1 x3 / 2 ( x  1)2

.626534929111853784164... 
   (2 k ) (2 k )  1
k 1

S2 ( 2 k , k ) 2 k

(2k )2 k
k 1
1/ 4
1 .626576561697785743211...  7
.626575479378116911542... 
.626657068657750125604... 
3 .626860407847018767668... 
1 
2 2
e2  e2
sinh 2 

2
2 2 k 1


k  0 ( 2 k  1)!


4k k

k  0 ( 2 k )!
AS 4.5.62

=
e
2 cos x
sin x dx
GR 3.915.1
0

.6269531324505805969...


k 1
 (3k )
2 1
k


(
k 1
1
3
2 )3
k
.627027495541456760072... 

 3k
3 log 3  4 log 2   3 
k 1
.627591004863777296758... 
3 .627598728468435701188... 
2
 1  2 
 1 2
        ,  
 3  3 
 3 3
3
dx
  x 2  x  1 
2

 
=
1
k /2
 ( 2 )   6)

=
2

3k


k 1  2k 
 k
k

log(1  x 3 )
0 x 3 dx 
dx
 1  cos x
0

=
ex / 3
 ex  1 dx


=
 log1  x  dx
3
0

.627716860485152532642... 
.6278364236143983844...

 


4
 log1  x( x  2)  dx

0
 (1)
k 1
 ( 3k  1)  1
k
k 1
.627815041275931813278... 





1
3

 k log 1  k

k 2

1
1

2e   erfi1  
4
k  0 k!( 2k  1)


1
4 4 5
1

arcsinh 
5
25
2

( 1) k
 1 1
= 2 F1  11
, , ,   
 2 4
k 0  2k 
 
k

.628094784476264027477... 
 sin 1

  (  2) cos1  log( 4 sin 2 1) sin 1 
 2 


 
=
sin 2 k


k  1 k ( k  1)

4 .62815959554093703767... 
.628318530717958647693... 
7 .628383212379538395441... 
2 (3) 


11 2
19


36
24

H k H k 3
 k (k  1)
k 1
3/ 2
x dx
 
5
1  x5
0
  i 3 

sin ( 1)1/ 3   sin 
2 
2
 1
 1   dx
 cosh 2   4 
2 

 x  x
  sinh x
=
cosh

1 .628473712901584447056... 
k
k 1
.628507443651989153506... 
.628527924724310085412... 

 
=
 3

2


3
  1  (2k  1)
k 0
2



1
k 1
Li3 ( ) 
(1)1/ 4


k
k
k 1
3
1
cot (1)    i cot (1)    

4
2
3/ 4
1/ 4
(1)1/ 4
  2  (1) 3/ 4     2  (1) 3/ 4 
4
( 1) 3/ 4

  2  ( 1)1/ 4     2  ( 1)1/ 4 
4


k2
   (2k ) 
k 1
(1)  ( 4 k  2 )  1   4
  Im  k 

k 1
k 2 k  1
 k 1 i 



=


5 2
14 .628784140560320753162...  2 ( 3) 
 4   ( k  1) 2  ( k )  1 
6
k 2
1/ k

k 1 e
2 .62880133541162176449...   ( 1)
k4
k 1

 
1
 0 (k )
.628952086030043489235...   k 2    k 2 j 2
k 1 2
j 1 k 1 2

106 .629190515800789928381... 
 (x
24 2 
0
4
dx
 1 / 16) 2
k 2 ( k  1)

2 e  1  2 e Ei (1)  
k!
k 1

5 .629258381564478619403... 
1 .6293779129886049518...

 

=
1 .629629629629629629629 =
.63017177563315160855... 


2
log 2 2

  log 2 

2 12
2
2
(  log 2 ) 2


12
12
44

27


k
4
k 1

log 2 xdx
0 e2 x 

e
 2x
log 2 x dx 
0
3
k
G

1
 (k )
.630330700753906311477... 
log 2     


2
k  2 k ( k  1)

1
1


 
=   ( 2 k  2 ) log 1    1 


k
2k 
k 1 
9 k 2  11k

3
k  2 k ( k  1)

.63038440599136615460... 

1
p 

 1  2
p prime
1
36
 1 4
2
( 2 )

4
.630628331731022856828...   ( 3) 
7
.6304246388313639332...
 1 
3 .630824551655960931498... 
2
e
log 3 2 

log  (k )
.631010924888487788268...  
k 1
k 2
.6309297535714574371...


sin 3  3k 
3 sin 1
 
4
3k
k 0

1
.631526898981705152108...   k
k 1 2  1 ( k )
12
.631578947368421052
=
19
105 
 9
11 .631728396567448929144... 
  
 2
16
.631103238605921226958... 
Berndt ch. 31
1
1 .6318696084180513481...

e
sin x
dx 
0
.6319660112501051518...

.631966197838167906662... 

9
1

  
4
k 1 F32 k
 (3) 
2
12
1
.632120558828557678405...  1 

e
 
=
Hk
k 2
k
k 1
( 1) k 1
 k! 
e

1
1

x2
Berndt 9.12.1
1
 ( x  1)e
0

2
1 .632526919438152844773... 


2
 1  dx
0 x cosh x dx  1 cosh x  x3
1

log 2 
GKP p. 302
2 


 ( k  1)
1 
1
log 1  
.6328153188403884421...  



k
 3k 
3 k
k 1
k 1 k

log 2 k   2  1 
.632843018043786287416...  

2k
2k
k 2
k 1 2
7
1096 .633158428458599263720...  e 
6 .63324958071079969823... 
44
x
log x dx GR 4.351.1
.633255651314820034552... 
 cos( e ) 

7 (3)
2 .6338893027985365459...   log 2 

2
2
.633974596215561353236... 
.634028701498111859376... 
.634340223870549172483... 

4k

k 1  2k 
  k 3
k 
3
1 3
1

1 

 (1) 
k
k

k 0
1
5
1 





 log   3  5   log   3  5   
 2 
 2 








 
=
F
 k  (k )  1 
k
k 2

 1  e2 
log sin 2 x
2 .634415941277498655611...    log
dx
   2
x 1
 2 
0
6 .63446599048208274358... 
Ei ( e)    1 
.634861099338284456921... 
2   csch
.63486687113357064562... 
arcsec
1 .634967033424113218236... 
.635181422730739085012... 

2

.635469911917901624599... 
.6356874793986674953...

.63575432737726430215... 
(1) k 1

2
k 1 k  1 / 4


 1
  x log log  dx    e 2 x log x dx
 x
0
0
2
k ( k  1)
 2G  

4
4k
k 1



k 2

 1 
2 ( 2 )  2 ( 3) 
 ( 1)
k 2

k 1
k

1
k
 2
3
4
k  2 ( k  1)
k ( k  1)( ( k )  1)
GR 4.325.8
4
 (4 k  1)
 (2k ) 

 (k ) 

=
ek

k 1 k ! k
15  33

Associated with a sailing curve. Mel1 p.153
6


 2 13
2k 2  1

  ( k  1) ( 2 k )  1   2
2 
12 16
k 1
k  2 ( k  1)

log 2 
   (1) k (k ) 
2
2 k 1
1
=

2
.635861728156068555366... 
1 .635886323999068092858... 
I1
 2

2

k
 (k !)
k 1
2
2k
1 
1 
1 
1 
1 

1
1
  1 
   1 
  
  
 
 2 2
2 2 2
 2 2 2 
4  2 2
( 1) k 1 k
 
=  2
k 1 k  1 / 2
1 
.636014527491066581475... 
 csch  
2 2


(1) k


2
k 0 k  1




 
.63636363636363636363

cos 2 x
0 cosh 2 x dx
=
7

11
6 .636467172351030914783... 

F3k 1
k
k 1 6

2  1

1  log 2 2
2

2
x
sin x dx
0


1
sin k / 2
.636534159241417807499... 


 
12 16 96
k3
k 1
i
Li  e i / 2   Li3  ei / 2 
=
2 3
.636584789466303609633...   ( 2 )   ( 7 )

GR 1.443.5

2
 2k 
1
.6366197723675813431... 

   
k

k  0  k  16 ( 2k  2)

1 

.6366197723675813431...   1 

4k 2 
k 1 
2

=
 cos 2
k 1
=
=
.636823470047540403172... 

  

GR 1.431

GR 1.439.1
k 1
 0 


1 / 2
i logi e
 1
1   1 
5  5   
 (1) 4 / 5   3  5  2i
 
 4
5
5 
2



1
1 
5  5  

2 /5  1





1
3
5
2
5
5
( 1)1/ 5   3  5  2i
(
)

i
(




4



5 
2
4









( 1)1/ 5  1

  
  3  5  i 2(5  5  
4

5


k3
=  ( 1)  (5k  3)  1   5
k 1
k 2 k  1


1
1


 (2k )
.637160267117967246437... 
coth
   2
  (1) k 1
2k
2 2
2 2
k 1 2k  1
k 1

6 sin 2 1
sin 2 k
.637464900519471816199... 
  k
4 cos 2  5
k 1 2
.637752497381454492659... 
.637915805122426308543... 
.638000580642192011016... 
1 .6382270745053706475...

k 1
3

2 e2
.6387045287798183656...



 
=


cosh 2



1
d
k
2 
2
6

cos 4 k
k2
k 1
 4  
 3  9 log 3
24


 6k
k 1
2
GR 1.443.3
1

 4k
log(1  x )
dx
x5
0
1

cos 2 x
2 2 dx
)
 (1  x
 4k

2
2
k  0 ( 4 k  2 )!

 ( k ) ( k  1)  1


k
k 2
1
k 2
.638251240331360040453... 


6
.638961276313634801150... 
sin log 2  Im2i 
.63900012153645849366... 
1  5  5
     
3  3  3

x 4 dx
e
x3
0
1
1  i 
1  i

 2 
  2 



i
 (4k  2)  1
2 
2
.639343615032532580146... 
  ( 1) k 1
log

1  i 
1  i
2
2k  1

k 1
 2 

  2 

2 
2 

1
 
=  arctan 2
k
k 2

1
5

 arc cot (cot 1  2 csc1)(cos1  2)  sin 1log  cos1  
.639432937474838301769... 
5  4 cos1 
4


H sin k
 
=  k k
2
k 1
.639446070022506040443... 
 (3)   (4)   (2) 

.6397766840608015768...

3
k 1
.640000000000000000000 =
k
k
1
16

25

Fk Fk 1
4k
k 1

3i
i
Li3 ( e i )  Li3 ( ei    Li4 ( e3i )  Li4 ( e 3i ) 

8
8
3

sin k
 
= 
4
k 1 k
4
 log 2 
.640186152773388023916... 
3
7
 log 2 
1 .640186152773388023916... 
3
1
502
x log 2 x
 
.64031785796091597380...  16 (3) 
dx
27
1

x
0
.640022929731504395634... 


H ( 3) k
.640402269535703102846...   k
k  1 2 ( 2 k  1)
.64057590922154613385... 
k3  2
 
k3
k 2

1
2 (2 )1/ 3   21/ 3    (1) 2 / 3 21/ 3 
e
.64085908577047738232...  2 
2
2 3
 H (3)1 / 3 
.640995703458790509...
 27  12 (3) 
3 3

2 .641188148861605230604... 
6G 
 /2

 
=

0
.641274915080932047772... 
3 2 G 3 2
1  ( 3)  3 
 1 


     (3)    
4
8
256 
4
 4 
x3
dx
sin 3 x


dx
  2

2x  3
2 6
0

 3x
0
dx
2
2
log x
 1
dx
 2 Li3     
 3 0 x  3
24

 5
 1
.641398092702653551782... 

 hg    hg   
 6
 6
5
3


1
1  1 / 22k
1 .641632560655153866294...   k ( k  1)/ 2  
2 k 1 
k 1 1  1 / 2
k 1 2
10
1
.6413018960103079026...
2

5 .641895835477562869481... 

[Gauss] I Berndt 303
(1) k 1
.642024000514891118449...   k
k 1 2  k


.642051832501655779767... 
 (1)k 1
 (4k  2)  1
( 2 k  1)!

1
.642092615934330703006...  cot 1  1  2 2 2

k 1 k   1

(1) k B2 k 4k
= 
(2k )!
k 0

k 1
.642092615934330703006... 
1 .642188435222121136874... 
7 .642208445927342710179... 

1
 sin k
k 2
2

GKP eq. 6.88
cot1 
25  2 5
1

5
7
G

, Lebesgue constant
 3
 Li2   
 4
9
.6428571428571428571
=
14
2

1 
 dx
.643107628915133244616...   1 

(
)
x


1
.64276126883997887911... 
1 .64322179613925445451... 
 4  i 21/ 335/ 6  61/ 3  
1

1     ( 2  61/ 3 )   
3
2


1  4  i 21/ 335/ 6  61/ 3 
 

3 
2



=
 6  (3k  1)  1

e sin 1  1

2

6
 6)
k 1
k 2


 (k )  1

 1 1
   Li1/ 2    
.643318387340724531847...  
k k
k
k 2
k 2

3
3
.643501108793284386803...  2 arctan 2 
 arcsin  arctan  gd log 2 
2
5
4
9 .6436507609929549958... 
93
.6436776435894211956...
1 .64371804071094635289... 
k
e sin e 

 k (k
3
e
 log x sin log x dx
1

 (1)k 1
k 1
ek
( 2 k  1)!
.643767332889268748742... 
 log 2
G

8
i   1  i
 1  i
  Li2 
 
 Li2 
 2 
2  2 
 /4

 
=
 log(1  tan x ) dx

GR 4.227.11
0
log(1  x )
dx
1  x2
0
1
=

1
=
GR 4.291.10
arctan x
 x (1  x ) dx
GR 4.531.3
0
.643796887509858981341... 
.643907357064919657769... 
.64399516584922827668... 
   (1  e)
e
1

 7

tan

6 4 2
2
1

3
log 2  4 log 2

1 .64429465690499542033... 
 2
(k )
k 2
 
.644329968197302957804... 
1 .644801170454997028679... 

 4k
k 1
2
1
 4k  6

sin 2 x
0 2 x
 2
log j
 2
j  2 k 1
.644756611586665798596... 
 He
1
jk
G5

4 3

sin  3 


k 1


1
 1  k (k  4)  


(1) k

k 1 k ( 2k  3) log k

.6448805377924315...

.644934066848226436472... 
2
6
 1   (1) (2)   (2) 
1


2
k 1 k  k  1

=
k 1
2
2
1
(k  1)

  (2k )   (2k  2)
k 1

 (1) k  (k )   (k  1)   k  (k  2)  1
k
k 2

=
k
  (k )   (k  1) 
k 2

=

1
k
k 2

=

k 1
1
k
k 1
2


dx
x
1
2
J272, J348
1
=
x log x
dx
1 x
0


=
log x
dx
3
 x2
x
1
1
=
 1
x log x 
2  dx

  1  x  (1  x )
0


 
=
x
x dx

x
 1
2
6
 
=
GR 3.411.9
  (2)   (1) (1) 

1
k
k 1


GR 4.236.2
 e e
0
1 .644934066848226436472... 
GR 4.231.3
2


2 1/ k
 ( k  1)
k 1
2

Hk
 k (k  1) 
k 1


 
=
1
k 1  2k  2
  k
k 
3

=

k 1
CFG F17, K156
21k  8
3
 2k  3
  k
k 

=
 k  (k  1)  1 
k 1
1
=

 k  (k )   (k  1)
k 2
log x
dx
1 x
0

GR 4.231.2


 
=
log(1  x)

x
(
1

x
)
0

1
=
log 2 x
0 (1  x)2 dx 

log 2 x
1 (1  x) 2 

log 2 x
0 (1  x)3 dx 

log x
 x( x  1) dx
1
1
=

0
log(1  x 2 )
dx
x3
log(1  2 x  x 2 )
dx
0
x
1
=
GR 4.296.1

=
  log1  e x  dx
GR 4.223.2
0


 
=
e
x
0


 
=
e
x
dx 
1
dx
x
0
2 .644934066848226436472... 
3 .644934066848226436472... 
 

 
6


 k  ( k )  1
1 
k 2

2
 k  ( k )   ( k  1)  2 
2 
6
k 2
 3
3 log 3

6
2

1  (1) k 1
1
 2
=         

2
 2
 3
k 1 3k  k

1


=  log 1 
 dx
 ( x  1)( x  2) 
0
.64527561023483500704... 


1
2
.64572471826697712891... 
1 .645902515225396119354... 
.645903590137755193632... 
.64593247422930033624... 
=

 (k )
k 1
Fk

2 .645751311064590590502... 
 2 log 2 
7
3
sinh


 1
2

3


1 


  1  3k
k 1

2
 (1  2i)  (1  2i)    sinx x
4
e 1
0
1
(1  i )
(1  i )  

 5  2 coth   (1  i ) 2  coth
 cot

16 
2
2  


1 1
coth  
 ( 1) 3/ 4   2  ( 1)1/ 4   ( 1) 3/ 4  2  ( 1)1/ 4 
8
16 8
1
 ( 1)1/ 4   2  ( 1) 3/ 4   ( 1)1/ 4   2  ( 1) 3/ 4 
8


1
 (8k  6)  1



6
2
k 2 k  k
k 1


=



.646393061176817551567... 

 (k )
k 2
k

 (k )  1

 
1
1
cos x dx
 cos  sin    4
2
2
4 e
0 4x  1
1
.6464466094067262378...  1 

2 2
 2 log 3 log 4 2 log 2 2 1  1 
1
.646458473588902984359...   Li2 ( 3) 


 Li2   
3
18
3
3
3  4
.646435324626649699803... 


 
=
e
x
0
x
dx
3

11
24
1
k !( k  1)!

arcsinh   (1) k 1
25 25 5
2
( 2 k )!
k 1
2

1
k
 5 sin 1  3 cos1   (1) k
.646612001608765845265... 
4
( 2 k  2 )!
k 1


( 6k  4)  1
.6469934025023697224...  
   k 4 log(1  k 6 ) 
k
k 1
k 2
.646596291662721938667... 


1
( 1) k 1  ( 3k  1)




2
1
2k
k 1 2 k  k
k 1

.64704901239611528732... 


1 .64705860880273600822... 

 1 
k 1
.6470588235294117
=
sin k 

2k 
11
17

1 .647182343401733581306... 
3 .647562611124159771980... 
1 .647620736401059521105... 
.647719422669750427618... 
.647793574696319037017... 
.647859344852456910440... 
log x
dx
( x)
1
36
6
2 

 (2)


4

(e  1) 
e
x2
cosh 2 x dx
0
1
cos1
cos1 
sin 1
 sinh
 cosh
 sin 1 
 
2
2
2  
2 


1
1 log 1  2
 

2
2 2
cos
3
2

arcsinh x
dx
x3
1


k sin k
k 
k 1 k ! 2

1 .647918433002164537093... 
.648054273663885399575... 
.648231038121442835697... 
.648275480106659634482... 
32 .648388556215921310693... 

log( x  2)
dx
x2
1
3 log 3

2

sech1 
2
1

1 
e e
cos i

E2 k
 (2k )!
3
arctan 2 log 5
log x cos2 x
dx


 
5
5
20
ex
0
2
 
3
6 3 


999 
 dx
3
 1
 log1  x
0

1
e  
k 
k  0 k !2
1 .648721270700128146849... 
 

1
cos k
 Li2 ei  Li2 e  i  2 2
2
k 1 k
 

1
 (2k )!!
 i  i /
k 0
 5  1
 5  1
3
  2

log 2 

 2 
12 4
2





 ( 4k )
.64887655168558007488...  
k 1  ( 4k  2)
.648793417991217423864... 
.64907364028134509045... 

2


2 3

=
k
tanh
k
 3
2
4


1

4

  ( 6k  4)  1
1
k 1

3
1
.649185972973479437802...  9 log  3   k
2
k  0 3 ( k  2)


 (5k  3)  1
.64919269265578293912...  
   k 3 log(1  k 5 )
k
k 1
k 2
k 2

6
log 2 1
 
3
3
6
1

.64963693908006244413...  sin
2
.649481824343282643015... 
 /4


0
x dx
cos 4 x

2 .649970605053701527022... 

AS 4.5.66
k 0

G 
3
16

 2 log 2
4
1


1
arccos x arctan 2 x
dx
x2
Berndt Ch. 9
.65000000000000000000
=
13
20
ee  1
1 .650015797211168549834...  log

e

e k Bk
(1)

k !k
k 1
k
Berndt 5.8.5
k2
.65017457857712101250...  4320  1589e  
k 1 k!( k  6)

3 .65023786847473254748... 


k
 3
 3 log 2      ( k  1) 
 
2
k 1 4

3
 k (4 k  3)
k 1
2/3





 (1)1 / 3 log1  (11)/ 3   (1) 2 / 3 log1  (11)/ 3   





3 
2
2




2/3
2
1 


log1  1 / 3 
3
 2 

1
1
1 1 2
2 2 5

  ,1,   1 / 3 2 F1  , , ,  1   k
=
2
3  2 3
3 3 3

k  0 2 (3k  2)

i
sin k
Li2 (ei )  Li2 (e  i )  Im Li2 (e  i )   (1) k 1 2
.6503861069291288806... 
2
k
k 1

2
1
1 .650425758797542876025...  erfi 1 

 k  0 k!(2k  1)

1
.650604594983248538006...   kH k
k 1 2
.650311156858960038983... 
2
2/3
1/ 3

 

 

2



 

2  1  arcsin  cos

2


(1) k
= 4

3
k 1 27( 2k  1)  2( 2k  1)
.65064514228428650428... 



 /2
=

0
1
=

0

sin 2 x
dx 
1  cos 2 x
 /2

0
cos 2 x
dx 
1  sin 2 x
1 x
arctan( x 2 ) dx
2
x
2

 dx
 1
0


 ( 4 k  2)  1
1
sinh 
1
.650923199301856338885... 
  arctanh 2
 
log

k
(2k  1)
2
k 1
k 2

1 (2)  3
1
3
2 .6513166081688198157...       28 ( 3)    

3 3
 4
2
k  0 ( k  4)
=
.651473539678535823016... 

 log 1  4 x
1
2
15
5 35
log 

32
4 64

k 2 Hk

k
k 1 5
GR 4.538.2
1 .651496129472318798043... 
.651585210821386425593... 
log 2  
1/ 4
1/ 4
   sin 2    sinh 2  

 9/4 
 2  cos 21/ 4    cosh 21/ 4  


 
=
 2k
k 1
2

 (1)
k 1
 (4k  2)
2k
k 1

1
 k 2
1

6  9  8 (3)  2   4  120 (3)  168 (5) 
48

H H
 
=  k3 k
k
k 1

H H
7 (5)  2
1 .651942792704498623963... 

 (3)   k 3 k
2
6
k
k 1
2 .651707423218151760587... 

1 .652163247071347030652... 
7 (3)

16

1 .653164280286206248139... 

k 2
 /2
x
2
log tan x dx
0
 2 (k )
k2
log x
31 6
  log(1  x )
dx
x
1260
0
2 .653283344230149263312...   ( 2)   (7)
1
23 .65322619113844249836... 
.653301013632933874628... 
4 
2 2 16
i


Li3 e  4i  Li3 e 4i
3
3
2
  
     sink 4k

k 1
3
GR 1.443.5
  (k

.653344711643348759976... 
2

 2)  1
k 2
40 .65348145876833758993... 
.65353731412265158046... 
11 e  7  11
e
1
erf

2
2
k3

k 1 k!!


1 
5  5   9  5 i 5  5  


4  1  5  2i


20 
2   4
2
2 

 


5  5   9  5 i 5  5 
1 


 1  5  2i

2   4
2
2 
20 

 

1 
5  5   9  5 i 5  5 

 1  5  2i


20 
2   4
2
2 

 

1 
5  5   9  5 i 5  5 

   1  5  2i


20 
2   4
2
2 

 




1
 
=  2
   (5k  3)  1
3
k 2 k  k
k 1

(1) k 116k
.653643620863611914639...   cos 4  
(2k )!
k 0

(1) k 116k

(2k )!
k 1


1
k !  (5k  3)
.653743338243228533787...   ( 1)


2
k
3
2
k 1
k 1 2k  k


(4 k  2)  1
.65395324036928789113...  
   k 2 log(1  k 4 ) 
k
k 1
k 2
1 .653643620863611914639... 
1 .654028242523005382213... 
.6541138063191885708...

5 .65451711207746119382... 
 cosh cos1  sinh cos1 sin(1  sin 1) 
2 ( 3) 
7

4

k sin k
k!
k 1


 (1)k 1 k (k  1) (k )  1 
k 3
2 ( 3k 2  3k  1)

k 2 ( k  1) 3
k 2

e

k 1
 (3k  1)  1
k
2


 1
2

 k Li  k
2
k 2
3
( 1)  2 k 
  k  
7
k
k 0 3

.654653670707977143798... 
k
4 10
sin  10 
3
8
6 .655258980645659749465... 
 (3)
.65496649322273310466... 

1/

.65465261560532808892... 

2 sin 2 2 



k 1

1

 1  k (k  6) 

( 3k  1)( k  1)
16G  
 (k  2) 
4k
k 1

k!
(1) k 1 k

k
k 1
 (2)  
  c1
2


 (2 k )  1
 1
  Li3  2 

3
k 
k
k 1
k 2
 3
Li3  
5


k 2  (k )  1
1
  2 (k  1)e1 / k  k

k!
k 2
k 2 k

14 .65544950683550424087... 
.655831600867491587281... 
.655878071520253881077... 
.65593982609897974377... 
.656002513632980683235... 
1 .65648420247337889565... 

Adamchik (27)
Patterson Ex. A.4.2


coth 1
1
1
.656517642749665651818... 
  2 2
2
2 k 1 k 
4 log 2 2 log 3
.65660443319200020682... 

1 
3
3
9 .65662747460460224761...  6 log 5
.6568108991872036472...

1 .65685424949238019521... 
4    3 log 2 
H2 k 1
k
k 1 4
3
2 3
log

2
2 3
1
 log
0
(1  x) 2
dx
1  x6
 2k  1 1
 
k  8k
k 0 
 
32  4 2 
5 .65685424949238019521... 
3 log 2 

.656987523063179381245... 



4 24 

2 .656973685873328869567... 
J951

3
8
1

arcsin 3 x
 x 3 dx
1
 ( k ) log k
2k
i
i


.65777271445891870069...     1      1  
 2

2
k 2
.657926395608213776872... 

 
=
.65797362673929057459... 
=
 3  i 
55 4
i   3 i
  Li3 

  Li3 
 2  
2592
2   2 



sin k / 6

k3
k 1
2
 (4)

  (4)   (3)
15
 (2)
(1)  ( k )


k2
k 1


=

k 1
1 .6583741831710831673...

.65847232569963413649... 
 
=
8 .658585869689105532128... 
(1) k
  2k
p prime k  0 p

 (k )

2
log 2 (2 x)dx

 
 2 log 2  log 2  
ex
6
0
2
 2 log 2
4

7 (3)

8
2
arcsin x
dx
x
0

8 (4) 
HW Thm. 299
HW Thm. 300
k2
2
1


p prime
p2

p2  1
GR 1.443.5
4 4
45
1

0
x arccos2 x
dx
1  x2

3 
 5
 3
 2  
    
2
6
 2
 2
 /2
 tan 2 x sin 4 x
=   x e
dx
cos 2 x
0
2 .658680776358274040947... 

 

1. 658730029128491878509... 

GR 3.963.3
 (k  1)
2k

 (k  1)  1
1  1
.658759815490980624984...  
   Li4   
4
k
 k
k 1
k 2 k

1
1
.658951075727201947430... 
Ei (1)     

2
k 1 2 k ! k
k 1

.65905796753218002746... 
4
  ( 3) 


 ( k  1)
72
k3
k 1
3 .659370435304991975484... 
(cosh 2  sinh 2) 1  cosh e  cosh(1  e)  sinh e  sinh(1  e)  
=
ee (e  1)  1

e2

ek


k  0 k!( k  2)




1 


1   2 k  
k 1
  k 
 k  
   


(1) k  (k )
1
 1 / k

1 
e


k!
k
k 2
k 1 
 2k  2  (1) k
2
 k 
 32   
40
3
 k  8
k 1
k 1 1 / k 



1
1





log  1   1     

   k   k   
k 2


 
 1
 1 
 Lik    Lik     

n
 n 
k 2 n2 

1 .659688960215841425343... 
.659815254349999514864... 
.659863237109041309297... 
.65998309173598797224... 

 
=
.660161815846869573928... 


 
=

1 
1 
 
( p  1) 2 
p prime  3 
Twin primes constant C2


p prime  3
p ( p  2)

( p  1) 2
75  5 3 
1


.660349361999116217163...    1 

2
3k  k  1
k 1 
 1 2 1
.660398163397448309616... 
 
 Li2 ei   Li2 e  i  
4 8
12 4
8 .6602540378443864676...
1 2/ p

2
p prime  3 (1  1 / p )



sin 2 (k / 2)

k2
k 1

.66040364132111511419... 

1

8
coth 2  
4

k
k 1
2
1
4
J124

=
.660438500147765439828... 
sin x cos x dx
ex  1
0



2
x

9/4
0
dx
2
4
3

2
2
 csc

5
5
5 5 5
1 .660990476320476376040...   G 

log k
1 .6611554434943963640...   2
k 2 k  k  1

1
(1) k kBk
 
.66130311266153410544...  1 
(e  1) 2
k!
k 1
.660653199838824821037... 
809 .661456252670475428540... 

x2
0 x5  1 dx
cosh cosh 2  sinh cosh 2cosh sin 2


cosh 2k
k!
k 0

1
1

log 2 cos   log (1  ei / 2 )(1  e i / 2 ) 

4
2

k  1 cos k / 2
=  ( 1)
k
k 1
1
.661707182267176235156... 
4  17 2  6 3  21log(1  2 )  42 log(2  3 )  7 
105
.661566129312475701703... 


 

Robbins' constant, the avg. distance between points in the unit cube
1 3 log 3
.66197960825054113427... 


4
8
2 .662277128832675576187...   ( 2)   (6)

2 .662448289304721483984... 

log( x  2)
dx
x3
1

1
  (k!) 
k 1
.662743419349181580975... 
Laplace limit. Root of

 3
6 .662895311501290705419...    
k 1  k 
.66333702373429058707... 

4

1 x 2
1  1  x2
1
k
coth  
1
.663502138933028197136...  
k 1 F3 k
xe
1

8

k2


4
k 2 k  1

  (4 k  2)  1
k 1
1  1 i 
1  i 
 Li3 
  Li3 
  
2  3 
 3 
2
1 .663814745161414937366... 
 (3)
.663590714044308562872... 
cos k / 4
k3
k 1


(1) k k
4G  2   2
1 2
k 1 ( k  4 )

( 1) k
4G  
1
1
k  0 ( k  2 )( k  2 )

1
1
1
28 (3)   3  64    ( 2 )    64  
1 3
2
 4
k 1 ( k  4 )
1 ( 2)  1  
1
3
28 (3)         
3
2
 4  k  0 (k  14)

 2k 
93
1
    k
140
1  k  4 ( k  1)

1 .6638623767088760602...

3 .6638623767088760602...

.663869968768460166669... 
64 .663869968768460166669... 
.66428571428571428571
=
4 .66453259190400037192... 
cosh

2



2
 1  (2k  1)
k 0

1
 2 

 
0 F1  ;1;    J 0 
e

 e

3 
2
.66467019408956851024... 
  x 4 e  x dx 
8
0
.664602740333427442793... 
 /2

 
=

0
e
 tan 2 x
2



(1) k


2 k
k  0 ( k!) e

1  2 cos 2 x
dx
cos 6 x cot x
GR 3.964.3

2 
x log x
(
)
(
)
dx
 1   1    2    x x

 6

e
e

1


0


 ( 3k  1)  1
.66489428584555810021...  
   k log(1  k 3 )
k
k 1
k 2
2
2 .66514414269022518865...  2
Gelfond Schneider constant, known to be transcendental
5 5
2 
6 3

1 .66535148216552671854... 
  5 log 5  5 log
 2 1 
5 10 
5 5
5 


6k  (k )  1
36
=
 

k
5
k 2
k  2 5k (5k  6)
.66488023368106598117... 
.665393011603856317617... 
1 2
 2 
1
   2 3 log 3  2 (1)    2 (1)    
18 
 3
 3 

 
=
2 .665423390461749394547... 
.66544663410202726678... 

Hk
k 1
 k
k 
  2k 
8 2
 arctanh

.665738986368222001365... 

 (k 2 )
k2
k 1
.665773750028353863591... 
arctan
1 .665889619038593371592... 
5
.666003775286042277660... 

(k!)

B2   
 2  2

2
  k I  k   k 
0
k 2
 
1 1
 log 1     

p  p 
p prime 



p prime
1

p( p  1)

HW Thm. 430

.666169509211720802743... 
2




  2 arctan(1  2 )
3
dx


  x
x
4 2
2
 2
0 e e
1 .666081101809387342631... 
 
4
 ( 2k )  1
k 1


2

1 .666090786005217177154... 
1

1 e

 3 (k )
k 2
k3
2 k  ( k  2)
8 .666196488469696472590...  

k!
k 0

e2 / k

2
k 1 k

.666355847615437348637... 
2 2
.666366745392880526338... 
cos sin 1
.66653061575479140978... 
sin 1 si (1)  cos1 ci (1)   cos1
1

 
=
  log x sin(1  x ) dx
0
.666666666666666666666 =
2

3
 2k 
1
  k


k  0  k  4 ( k  1)(k  2)

F4 k  3

k
k 1 8

=
=
(1) k
 2k
(1) k 1

k 1 ( k  3 / 2)( k  1 / 2)



=
1
 1  (k  2)
2

2 

1  3 

k  1
k 2 
k 1




=
J1034, Mardsen p. 358
 /2
=
 sin x
cos x dx
0
 /2
=

0
1
dx
1  sin 2 x

=
cosh x
dx
e2 x
0


=
 cosh log x
1

=

x8 dx
ex
0
1 .666666666666666666666 =
dx

x3
3
5
3

F4 k 1
k
k 1 8
8
2 .666666666666666666666 =
3

F
=  4kk
k 1 8

1
2 .666691468254732970341...  

k 0 ( k!!)!
=

.666822076192281325682... 
12
.666995438304630068247... 
cosh 3 sinh 3


2
2 3
1 2 
 
.66724523794284173732...  2  log
 2 
.667457216028383771872... 
.66749054338776451533... 
arccos

2


( 1) k 1 ( k  12 )!

k !k
k 1

4


1
  2  e2i     2  e  2i  
4
  ( k  1)  1 sin
k 1
3k k

k  0 ( 2k  1)!


 

2
k
.6676914571896091767...

1 1 1
 1 3 
   ,     ,   
2  2 4
 2 4 

.66774488357928988201... 

 (2 k )  1
k
k 1
 

k 0
(1) k
2k  1
 1
2

 Li  k

k 2
2
 log 2
log 3 2
 1
 Li3    
6
6
 2


1
x 2 dx
log 2 x dx
log 2 x dx
= 
 8 3
  x
0 1  2x
1 x  2x
0 e 2
1 .66828336396657121205... 

2


 Li3 ( 2) 

.66942898781460986875... 
k
 2
k 1
96 .67012655640149635441... 

 o (k )
2
k 1

.669583085926878588096... 
2
k2
k
1
 1 (k  1)

2 log 2  8 log 2  12 (3) 
3
3


x 3 dx
0

1 .6701907046196043386...

1 .67040681796633972124... 
15 .67071684818578136541... 
20 .67085112019988011698... 
e
k 1
5

k
1
1
2
k 1

ex  1
k
k
   5
2 3
3


1
 1 
 e 1  e log1      k

 e   k  0 e (k  2)


(1) k
.671204652601083...
  2
3/ 2
k 2 k  k

(1) k 1 e1 / k

2 .671533636866672060559...  
k5
k 1
1

1 .67158702286436895171...  sin   sin 1cosh cos1  sinh cos1 
2

.670894551991274910198... 


1
 k! sin
k 0
2k  1
2
2
.671646710823367585219... 
e (1  erf 1)

2
.671719601885874542354... 
1

dx
2 
1
1 5
 log    log 2     log log 
2
3
 x 1  x  x
6 6

0
e x
0 1  x 2 dx
GR 4.325.6

=
.67177754230896957429... 
.671865985524009837878... 


=
log x
1
1
0 e e

x
10
1  1
 

27 3 9 k 1  2k 
 
k 
 (x
dx
 1)3
3
0

1

k
k 1

1 
1
  (1) k 1  (2k  1)  1     (4k  1)   (4k  1) 
2 k 1
2 k 1
 
1
 (i)   (i)  
2

k
3


1
Re  2

 k 1 k (k  i ) 

92 4 log 2
1
.672148922218710419133... 


 
75
5
k 1 k ( k  5 / 2)
=

1
.672247528206935894883... 

1
.672462966936363624928... 
.67270395832927634325... 
.672753015827581593636... 
.672942823879357247419... 
arcsin 2 x arctan 2 x
dx
x2
1
 Y0 (2)  2  J 0 (2)  
2
90
6
 30 (4)    (2)    (kk) log k

 4k
k 1
1
1 3
 log

4 3 3
3
 
3
(k )  2 (k )  1 
 
3
(k )   (k ) 
 
3
(k )  1 
k 2

4 .67301790986491692022... 
k 1

k 2

3 .67301790986491692022... 

2
1



cot

4 4 2
2

2 .67301790986491692022... 
(1) k 1 H k

k!k !
k 1

2
1
2
2
(1) k ( 3  1)3k 1

3k  1
k 0




k 2
1 .67302442726824453628... 
=
.6731729316883003802...

e 2 1 cos 2
cos 2  cosh 2
 cos(1  i ) cosh(1  i ) 



2
4 4e 2
2

16k


k 0 ( 4k )!
1  3


3/ 4    cos
 4
2
8
sin 2  x 4 
0 x 2 dx

2



1
ee
sinh 1

1
.67344457931634517503...  
k 1 q ( k )
2 .67323814048303015041... 

1 
sin 2 x
.673456768265772964153... 
  2si (2)   2 dx 
2 2
x
1
2

k
.6736020436407296815...   k

k 1 4  1



1

.673917363376357541664...   1 
2 
k (k  1) 
k 1 
2
3
5 1
 log
20 8
2


 (k )  1
1 1
.674012554268124725048...  

Li3  

3
k
k  2 ( k  1)
k 1 k
.673934654451819223753... 
Berndt 9.8
(3k  3)(1  (1) k )

8k !
k 1
2

k2
7 (3) 
.674600261330919653608... 

 6   k  (k  1)  1 
2
4
k 1 2

pk
3 .674643966011328778996...   k  pk the kth prime
k 1 2
3 .674225975055874294347... 
cosh 3 1 


1 .674707331811177969904... 
1
J844

2(2k  1)
 (2k  1)
k 2

 1  2 ( (k )  1) 
k 2

.67475715901610705842... 



k

 ( 2k )  1
F2 k
log k

2
k  2 ( k  1)
k 1

.67482388341871655835... 
.675198437911114341901... 

p prime
5 .675380154319224365966... 
.67546892106584099631... 
.67551085885603996302... 
1

p!
 (k ) k

 (k  1)!
k 1


6
  log 2  3 log(2  3 ) 
5
1

1  log 2 2
arctan 2

2
(1) k

k  0 ( k  1)( k  1 / 6)


sin x

x
0 2

  2 arctan
2 2
1
2 

 3x
0
2
dx
 2x  1
3


=
dx
0 x 2  2 x  3 
k  0 (k )


2k
k 1

14 .67591306671469341644... 
.675947298512314616619... 
2
0
dx
1  sin 2 x
m2n2


mn
n 1 m 1 2


log 5
arctan 2 

2 2
4 2
1 1 /(1
e
5
5)
e
5/2
 /4

0
cos x
dx 
1  sin 4 x
  kF!2
1 

k 2 2 k  4 k

m 3
)
m 1 (1  2

1/


0
2
1
dx 
1  x4

k
k 1
k

 0
e  e 
 2k
   

 

2
k  0 ( 2k  1)!
i





k 2  2k  2
1
2

   2
 
=  1  2    1 
k 
k ( k  2) 
k  2k
k 1 
k 1 
k 1


 ( 2k )  1
1
1
.676565136147966640827...  
  arctanh
k
2k  1
k 1
k 2 k
3 .676077910374977720696... 


1

.675977245872051077662... 
 /4
sinh 

1 .67658760238543093264... 

k 1

.676795322990996747892... 

k 1
1 .676974972518977020305... 
(k )
k!
(k )
2k k

1 

 
e1 / 8 1 
erf

2
2
2



.67697719835070493892... 
  ( k ) 
k 2
.677365284338832383751... 
12  2e 
.678097245096172464424... 
3 .678581614644620922501... 
17 .67887848183809818913... 
2 .67893853470774763366... 
1

k
k  0 k!!2
(k  1)!!
k 1 ( 2k )!!


  ( k ) 
16
 12  18 cosh 1  14 sinh 1 
e
3 
  ( k )   ( k  1) 

 
 1

2 12 k  2 
2
3

3 1
sin k
   3 
8
2
k 1 k
2

 (k )

k  2 ( k  1)!

( 2 k )!! 2 k
e 2 2  erf 2  
k 1 ( 2 k  1)!! k !
 1
 
 3
2
.67753296657588678176... 
(k )



1
 (2k )!(k  2)
k 0
.678953692032834648496... 
.67910608050053922751... 
cot 1
2



1
K ( 2) 

1  2  
4
e 
4
2 1/ 2
1

sin 2 x dx
0 1  x 2 
 /2

 
=
 sin
2
(tan x) dx
GR 3.716.9
0
719 .67924568118873483737... 
61 7
 360i Li7 (i )  Li7 (i )  
256

.679270890415917681932... 

 2
j  2 k 1
2 .67953572363023946445... 
1
  
2
2
log 6 x
0 1  x 2 dx 
GR 4.265

4
cosh  sinh  csch  3 
3

.679548341429408583377... 
1
jk
1
k2  4

2
k 0 k  3


(k )  1
k 2

.67957045711476130884... 

e  k2
dx
1


  x
3
4 k 1 (2k )! k  0 4k!
0 e ( x  2)
H k  (k )  1

k
k 2

.679658857487206163727... 



2 .679845569858706852438... 
11 .680000000000000000000 =
log k
k 1 Fk

292
25


1 .680531222042836769403... 
k 2 F2 k
k 1
4k

(1) k  (k )


k  2 k ( k  1)

.680531222042836769403... 


 H
k 2
k
 1 (k )  1

k
k
 H  (k )  1   k  1 log k  1
k 2
k
k 2

7 .6811457478686081758...

59 
1
 cot  
1

 2
2
2
k 1 k   1

dx
 1
.681690113816209328462... 
  2

x
1
.68117691028158186735... 
81 / 4
3
1 i
.681942519346374694769... 
 2 F1 1,1,1  i,  1 
2 F1 1,1, 2  i,  1 
2
4
1 .681792830507429086062... 


k 1
k2
2k (k 2  1)
.68202081730847307059... 
 log 2
G
4

.682153502605238066761... 
3
k 1
k
3
16

3
192

 0 (k )
k 1
3

k

 /4

x


2
cot 2 x dx
0
1
 3 
j k
j 1 k 1

Berndt 6.1.4
k2
(3k  1)

1

  1   
k! 
k 1 
k 1
3 .6821541361836286282...
2
3k  1

=
1

1

7 4
21 (4)
log 3 x
dx 

 
2
x
120
4
1

0
1
5 .68219697698347550546... 


x3
log 3 x
dx
1 x 2  x  1 1  e x dx 
GR 4.262.1
.682245131127166286554... 
1 
 


1
 2  2  2  2
2
   2  (2k  1)  1

k
k 1

.68256945033085777154... 
2 sinh ( / 2)
.682606194485985295135...  log 5 3
.6826378970268436894...

720
 52
cos

143
2
.682784063255295681467... 

.682864049225884024308... 
  (k


k 1

1

 1  k (k  7) 

1 / 3

2

 1)  1 
k 1

.68286953823459171815... 

 
=


k 2
1 .682933835880662358359... 

1
 21  4 2 36 coth   12 2 csc h 2 
192

1

k 2 1  1/ k 4
cosh

2 2


2


1
 1  2(2k  1)
2




i
1 .68294196961579301331...  2 sin 1   i (e  e )
1
1
1

1 .683134192795736841802...   cosh cosh  sinh cosh  sinh sinh

2
2
2

1 e
sinh( k / 2)
=

e  e1/ e  
2
k!
k 0
k 0
i


7 2  3
9
1
 2

log 3  log 2 3   (1)   
24
4
8
2
 3

1
1 

.683853674822667022202...     log1  sin  
k 

k 1  k
.683150338229591228672... 
J712

Hk
2
k
 3k
k 1
3 .683871510540411993356... 
Ei (2)  log 2   
1
2k
1  e2 x



0 x dx 
k 1 k!k

( k )

k
k 1 2

.683899300784123200209... 

.68390168402925265143... 
1 
3
 1 
192 2 G  2 4   ( 3)     ( 3)    
32 
4
 4 

1

 
=

0
.6840280390118235871...

arcsin 4 x
dx
x2
2
6
( 2 k  1)!!


2
k 1 ( 2 k )! k

 2 log 2 2 

( 2 k )!

2 k
4 k2
 ( k !)
k 1


 

 
.684210526315789473
.6842280752259962142...
Hk
= 

k 1 ( k  1)( 2 k  1)
1
log 2 (1  x )
= 
dx
x2
0
13
=
19

( k )  1
 
k  2 2 ( k )
21 .68464289811076878215... 
.6847247885631571233...

 


=
=
=
2
3

7 4
 9 (3) 
90


log K0 log 2 
1
log 4 x
0 (1  x)4 dx
 (2 k )  1  2 k 1
1
  ( 1) j 1  
j
 j 1
k
k 1
 K0 Khintchine's constant
1 1 
 4
   (1) k Li2  2 
k 
2 2 k 2

1
 1 
 ( 2 )  log 2 2   Li2  2 
 k  1
2
k 2

 1  1
  log1   log1  
 k  k
k 2

log 2 2  Li2  

Bailey, Borwein, Crandall, Math. Comp. 66, 217 (1997) 417-431

.68497723153155817271... 

(  1)
2
2 k 1


k
k 1 k

1 .685290077416171813048... 


k 1
k

k


 (k )

k 1
k
1
1
x
0
 2x
dx
Prud. 2.3.18.1
1 .68534515180093406468... 


2
sec
 5
2



 1  k

k 1
log 2 k

1
2 .68545200106530644531... 
  1  k (k  2) 
1 .685750354812596042871... 
1
K  
 4
3 .685757453295764112754... 
5 2

  (3) 
4
8
2
1


 3k  1 
 Khintchine's constant


.686320834104453573562... 

H k H k 2
 k ( k  2)
k 1
  (k )  1
k 1
k 2
.68650334233862388596... 

 
=

 
=
.686827337720053882161... 

 5  i 3 
5i 3
1 

  1 i 3  
2  1  i 3  
 2  

6 
2






1
1


(1) k 1  (3k )  1


3
2 k 1
k 1 k  1


1
 (3k  1)  1



1
2
k 2 k  k
k 1


tanh
 7
7
4 (3)
.686889658948339591657... 
7
3G
.6869741956329142613... 

4
1 .687023683370069268005... 
2G 
.687223392972767270914... 
7

32

.687438441825156070767... 

2
8



1

2

2

 log 2
2
k
k 1
2

1
k2
1
arctan 2 x
 x 2 dx
1


cos 2 x sin 3 x
0 x3 dx
 (k )  1
k2  3
1 .687832499411264734869...  3  ( 2)   ( 4) 
k 2
.687858934422874767598... 
.688498165926576719332... 

3 
 3 tanh  tanh  
8
2
2
 (2  2 )
2

.6885375371203397155...

9/4
1

   1  k 
4 
k 1 

sin 3 x
 e x  e x dx



e
0
x2
cos( x 2 ) dx 
1 .688761186655448144358... 
.688948447698738204055... 
.689164726331283279349... 
1  2   32    (2k 11)

3 / 2
 
I 2 ( 2) 

3/ 2
1
 k!(k  2)!
k 0

 ( 2k )  1
k 1
k!

k 0

LY 6.114
 e

1/ k2
k 2

1 

3 .689357624731389088942... 
.689723263693965085215... 
 (log 2  arccsch 2) 
3
6
 2 log 2  2 log 2   
2


x 2e  x
ex  1
0
3 5
2
2
1
1
log
 log 5 

5
2
5
5
.68976567855631552097... 
1
.690107091374539952004... 
arccsc
.69019422352157148739... 
log( x 2  5)
0 x 2  1 dx
2



1/ 4

(1) k 4 k  (k )

5k
k 2

7 .690286020676767839767... 
2e
7 log 3
4 .69041575982342955457... 
22

e
x2
cos x dx
0

.69054892277090786489... 
.690598923241496941963... 
.69063902436834890724... 
cosh 2 1 1
 
2
2
3
4

3

1
 sinh x cosh x dx
0
 2k 
1
  k  6 (k  2)
k 0


k

1
cos 3k
log 2  2 cos 3   

2
k
k 1
1
3  1
log 3 x dx
 1 
  Li2     Li3       
3
4  2
 2 
0 ( x  2)

(1) k 1 k
1
.69088664533801811203... 
(cos1  sin 1)  
2
k  1 ( 2 k  1)!


1
1


cot
  2
  2
2 .69101206331032637454...  1 

2
2
k 1 k  1 / 2
k  0 k  2k  1 / 2

 ( 2k )
 
=  k 1
k 1 2
.690759038686907307796... 

.691229843692084262883... 

2
 /2
J1 (1) 
 cos(sin x) cos2 x dx 
0
 /2
 cos(sin x) sin
0
2
x dx 
 /2
=
 sin(sin x) sin x dx
0
4 .69135802469135802469... 
380

81
k4


k
k 1 4


.691367339036293350533... 
I 0 ( 2)

.69149167744632896047...  1 
e2
.691849430120897679...

 3
  
  4
1
4k  1 

 1  4k  3 
k 0
( 1) k 1 (2 k )!


( k !) 3
k 1

sin( 2 ) sinh( 2 )

 
6 2
.691893583207303682497... 
4  2 log 2  4 log 2 2 
.69203384606099223896... 
 ( 2 k  1)

( 2 k )
k 1


4
4 

 1  k
2

kH k
 2 ( k  2)
k 1
k



3 (3) log3 2


2
3

(1) k
1 / e
 
.692200627555346353865...  e
k
k  0 k!e
9
.692307692307692307
=
13

1
 .6926058146742493275...   2

2
k  2 k log k
1 .692077137409748268193... 


.6929843466740731589...

=


( 2)
Hk H
k 1
2k

  3  5 
3 5 
1 
1

   2  5 Li 2 
4 Li 2    2  5 Li 2 



5 
8
8
4




k 1

(1) Fk F3


4k k 2
k 1





3 2 G
1  ( 3)  3 
 1 

     (3)    
2
128 
4
 4 

(1) k 1
.69314718055994530941...  log 2   (1)  
k
k 1

1
1
=  k  Li1  
 2
k 1 2 k
1 .6929924684136012447...

 /2

0
x 3 dx
sin x
J71
J117
=
2 arcsinh


 
=

 (k )
1
2 2
 2 arctanh

1
1
 2  2 k 1
3
(2k  1)
k 0 3
K148

2k

(2 k )  1


K Ex. 124(d)
log(1  k 2 )


k
k 1
k 2


(2k  1) (2k  1)
(4 k  1) (2k )



2 2 k 1
2  4 2 k 1 k
k 1
k 1

1
GR 8.373.1

k  0 ( 2k  1)( 2k  2)



(2k  2)!
(k  1)!
(2k  1)!!


J628, K Ex. 108b



(2k )!
k 1
k 1 ( 2k )!!
k 1 ( 2k )!!( 2k )
1 
1
 3

[Ramanjuan] Berndt Ch. 2
2 k 1 8k  2k

(1) k
1  2 3

[Ramanujan] Berndt Ch. 2
k 1 8k  2k
3 3  (1) k
 

[Ramanujan] Berndt Ch. 2
4 2 k 1 27 k 3  3k

Hk

2
k 1 4k  1
k 2


 
=
=
=
=
 
=

 
=

 
=
=


H (n) k 


1



k  

n 1 
k 1 2

=
cos k 
k
k 1
n
1
= lim 
n 
k 1 n  k
 
1
=  

r
r  2 s 1 ( 2 s )



 
 
=


=
dx
2 x 2  x 


 
=
e
0


 
=

 
x
1
[Ramanjuan] Berndt Ch. 2
J1123
dx
x
3
dx

1
x
e  x dx
e



GR 1.448.2
e x
1

dx
= 

2
2 x  3x  1
0

x
0
2
dx

 3x  1

1

 
=
log x
log x
dx  
dx
2
2
0 ( x  1)
1 ( x  1)


=
log x
 (1  x )
GR 4.231.6
GR 4.234.1
dx
2 2
1
1

 
log(1  x 4 )
=  
dx 
x3
0

 
=
 /2

0


 
=

 
=
x
dx 
1  sin x
 /4

0
dx

(cos x  sin x) cos x
4
sin x
dx
x3
0


x
 cosh
2
0
x
dx 

 /2

 
=
 sin( x) log tan x dx
 /2
 
0
 cos( x) log tan x dx
GR 4.393.1
0
1
=
  log(sin  x) cos 2 x dx
GR 4.384.3
0

=
dx
 (1  x ) cosh( x / 2)
2
0

=
tanh( x / 2)
dx
x
x
cosh
0

GR 3.527.15
1
=
  li ( x) dx
GR 6.211
0
2 .69314718055994530941... 
2  log 2 


 
=

1
log( 2 x ) log x
dx
x2
 2 
J0
 
 3

 (k )
14 .69353284726928223312...   3
k!
k 1
.69384607460674271193...    (3)
.693436788179183190098... 
5 .69398194001564144374... 
2k  1

k
k 1 2 k


2
3
(1) k

2 k
k  0 ( k!) 3

 2 (3) 


k 2
k 2
2k
 k (k  1) (k )  1   (k  1)
3



 
=

 (i  j ) (i  j )  1
i 1 j 1
.694173022150715234759... 
=
.694222402351994931745... 
.694627992246826153124... 
1 .69476403337322634865... 
 

k
k 1
(1) k 1 4k

k 1 ( 2k )!k

log k
2
 2k
 6
9 .695359714832658028149... 
94
.69568423498636049904... 
k !k
 (2k )!
2 log 2  ci(2)    
7 .69529898097118457326... 
8 .69558909395047469286... 

1 / 
k 2


 
1 3e1 / 4 
1
erf 

4
4
2

.69515651214315706410... 
 
8  6  6 log 2 

 (k  1)k 2
k 1
2k



i
  2  ei     2  e i  
2

 

  ( k  1)  1 sin k
k 1
1 .696310705168963032458... 
 e
 
2 sinh 2 

 2 
2 .696310705168963032458... 
cosh e 

1 .696711837754173003159... 
 

i
 1  (1)1 / 4   1  (1)1 / 4   1  (1)3 / 4   1  (1)3 / 4 
4


1
k


(1) k 1  (4k  1)  1


4

1
2
k
k 1
k 1

ek

k 1 ( 2k )!

ek

k  0 ( 2k )!
u(k )
k
1
2
k 1

( 1)
Ei (1)
 
.69717488323506606877... 
e
k 1

1
.697276598391672325965...   k
k 1 2  k
 2 
16

 
1 .697652726313550248201... 
3
1 / 2 
k 1
 ( k  1)  e k  
k!

log 2 x
1 3x 2  x dx
 1
 2 Li3   
 3
.6976557223096801684...
.697712084144029080157... 

1
1  I 0 (1)  L0 (1)    e x arccos x dx 
2
0
.69777465796400798203... 
I 1 ( 2)

I 0 ( 2)
Borwein-Devlin, p. 35

has continued fraction {0, 1, 2, 3, 4, 5, …}
.69778275792395754630... 
.69795791415052950187... 

.698035844212232131825... 

2
1  1
 Li2    
18 3  2 

4 2

csc


2

1
log x
 ( x  2)( x  1) dx
0

 1  2k
k 2

1
2


 1

625


 3 5 log 5  5  log 25  11 5  3 5 log 2  
 log
2
5 3 5 


FF

 
=  k k k  2
k 1 4 k


1
1
   (1) k 1  (5k  2)  1
.698098558623443139815...   5
2 k 1
k 1 k  1

1


7 .69834350925012958345... 
sinh 4

2 

4k


1
k  0 k !( k  2 )


 .6984559986366083598...

sin 2  (1) k 2k


2
k  0 ( 2k  1)!
=
(1) k 1 2k k

(2k )!
k 0
GR 1.411.1

.699524263050534905034... 
8    6 log 2 
.699571673835753441499... 
|  ( k )|

k
k 1 2  1


 2k
k 1
2
1
 k /2
.70000000000000000000
7
=

10
.700078628972919596975... 
=
.700170370211818344953... 
1  p 2  p 4

2 2
p prime (1  p )
1
 1

 (1  i )   (1  i )  (1  (1) 2 / 3  
4  4(1)1 / 3  
6
 2






2 .700217494355001975743... 
.700946934031977688857... 



 k  (k )  1
3
3
k 2

.700367730879139217733... 



1
1

   1  (1)1 / 3   1  (1) 2 / 3  1  (1) 21 / 3  
4  4(1)1 / 3  
6
2

1
  1  (1)1 / 3  1  (1) 2 / 3
6


1
1
   (1) k 1  (6k  3)  1

3
3
2 k 1
k 1 k  k



 (2k  1)
 1  1
 k Li2  2   



2
k  k
k 1 ( k  1)
k 1 
1 

2k

k 1

S 2( 2 k , k )

k 1 ( 2k )!k

 1  2

1 .70143108440857635328... 
x 2 dx
0 e x  x

.7020569031595942854...

.70240403097228566812... 
=
.70248147310407263932... 
.702557458599743706303... 

dx
1
1
 (3)    3   3
x
2
k 1 k
1
1  1 i 
1 i 

  (1  i )  
  (1  i )
2
 4 
 4 

1

2
1
k 1 k  k  1  k

2 5


x
0
42 e 
dx
5
2

1
 k ! 2 ( k  2)
k
k 0
.703156640645243187226... 
1 .70321207674618230826... 
.703414556873647626384... 
5
 2
  

8
   2 log 2
3

k
 2
 log 3      (k ) 
3 3
k 2  3 
arctanh
1
1
1
  log tanh 
4
2
e

 9k
k 1

e
k 0
2
4
 6k
1
(2k  1)
( 2 k 1) / 2
J945
1
1
1

  log 2 sin  
log
4
2(1  cos(1 / 2))
2


cos(k / 2)
1
=
 log (1  ei / 2 )(1  e  i / 2  
k
2
k 1
k

2
2 .703588859282703332127...  2 HypPFQ1,1,1, 2,2,2,2  
2
k 1 k!k


 (k )  1
2 
1
   log1  k 
.703734447652443020827...   k  2
(k  1)
 2 
k 2 2
k 2 k
.703585635137844663429... 

GR 1.441.2

G4

dx
 .7041699604374744600...   x
x
1
.703909203233587508431... 

.7044422...

48 .70454551700121861822... 


1
1 

p ( p  1) 
p prime 

4
2



 (k )  1
1 
(k )

  1 
  

 (k ) 
k 2  (k )
k 2 
k  2 k ( k  1)

 (k ) 
1
  2  

.70521114010536776429... 

 
=
k

1
=  k
m 1 k  2 m  ( k )

(k )
1 .70521114010536776429...   
k 2 k  1

1
.7052301717918009651...   
, p(k) = product of the first k primes
k 1 p ( k )
k 1
 
k

.70535942170114943096... 
 exp( ( x)  1)  1 dx
2
.70556922540701800526... 
315
sin  17 
13 17


1

 1  k (k  8) 


1 
1
k  1  ( k  1)
.70561856485887755343...   ( 1)
  log 1  
k

2 k
2k 
k 1
k 1 k

(1) k
.70566805723127543830...  2 J 2 (2)  2
k  0 k!( k  2)!

2 .70580808427784547879... 
4
36
k 1


 0 (k )
k 1
k2
  2 ( 2)  

Titchmarsh 1.2.2
=

 ( 2) (k )
k 1
k2

12
17

 3 
1


.7060568368957990482...   
 1   log 1 
 dx
2
 2( x  1) 
 2 
0
137
1 .70611766843180047273... 
    (6)
60
.7058823529411764
=

.70615978322619246687... 
 2  (k )  1) 
4
k
k 2

.70673879819458419449... 
   (k  1) (k )  1 
k 2
.7071067811865475244...

 
2
1




 cos  cos  sin 
2
2
4
12
12


 2k 
1  2k 
k 1
=  ( 1) k     k   
4  k  k 0 8  k 
k 0

( 2 k  1)!! k
= 

k
k 1 ( 2 k )!! 2


(1) k 
 
=  1 
6k  3 
k 1 


( 1) k 

=   1 
2 k  1
k 1 


 

 
1 .707291400824076944916... 
.707355302630645936751... 
 3 (3)

 (6)
20

9



(1) k 
1 


6k  3 
k 0 
 0 (k 2 )
k 1
k
3


J1029

lc(k )

3
k 1 k

Titchmarsh 1.2.9
2
 2k 
1
 

k
k  2  k  16 ( k  2)


k
1
  1  log 2   
 ( k )  1

2
k 2 k  1
15 .707963267948966192313...  5


 (k  1)  1
1 1

.70796428934331696271...  
Li2  

2
k
k
k 1
k 2 k
.70754636565543917208... 
.70796587673887025351... 
.7080734182735711935...

7
7
 
1 
2
11 
sin 2 1
 2 k  (1) k
  k

7
k 0  k 

1  cos1  (1) k 1 22 k 1

 H ( 2 )1 / 2 
2
(2k )!
k 1
GR 1.412.1
.70807785056045689679... 
 2  i
 2  i
 4  i
 4  i
4 log 2   
 
 
 

 4 
 4 
 4 
 4 
( 1) k 1

2
k  1 k ( k  1 / 4)


 
=
45  3 5
6 .70820393249936908923... 
.70828861415331154324... 
.70849832150917037332... 
1  1 1
 1 2 
   ,     ,   
6   2 6
 2 3 
(1) k

3k  1


k 0
(1) k 1
2k  1
sin

k
2
k 1

1
1
1
 (cos  log2  2 cos1sin  
2
2
2

.70865675970942402651... 
   (k ) log  (k )
k 2
.70869912400316624812... 
231

1024
1

0
x11 / 2
dx
1 x
GR 3.226.2

.70871400293733645959... 
.70875960579054778797... 
1
k
k 2 k
1 
2
8
 4G  2 


 
=

e

2
1 .70953870969937540242... 
.70978106809019450153... 
1 .709975946676696989353... 
.710131866303547127055... 
Hk
2
k
 4k
k 1
4 .7093001693271033307...
3(  8)
9
log 2  log 2 2 
2
2


8 2
3 
  ,0,1 
 2 
e


2
=

1 .710249833905893368351... 
5
4  2 (2) 

2
 k (k  1)
 log( x ) log(1  x)dx 
2
sinh 2   16 
  1  4 
120
k 
k 3 

1
k
k 1
1
2
)!

2
 
2 
0
1/ 3
0
.7102153895707254988...
k 0
 (1  i )e( 1 i )
1
 
k
 (k 

(1  i ) e( 1 i )
k 1


k2 1/ 9
2

cos2 x
dx
1  x4
8 .71034436121440852200... 
4 log 2 


x2
0 1  cos x dx
GR 3.791.6
1
1


 ( 3k 2  k ) / 2  ( 3k 2  k ) / 2 
2
2

k 1

1

= 1   1  k 
2 
k 1 

2k

 (k  1)  1
.711392167549233569697...  1 
 
2
k 1 k  1
.71121190491339757872... 
 (1)
3 .711471966428861955824... 
1 
k 1
Hall Thm. 4.1.3
3 5 
1 
2  
  2  2 
 (7  3 5 ) 
 

5  3 5 
1  5  

 2 


 
=
 F  (k )  1
k 1
k 2
.711566197550572432097... 
.711662976265709412933... 
.71181724366742530540... 
MHS (2,1,2) 
9 (5)
 2 (2) (3)
2
3 
 coth 
2 4

 13
4

tan
3
2
13

1 .71206833444346651034... 
  0 (k ) (k ) (k )  1
1 .71219946584900048341... 
  2 ( 2k ) 

 1 

k 1   ( 4k )

1 .71231792754821907256... 
(1) k
1  log 3  log 4  1   k
k 1 3 k
k 2

2 ( k )


2s
s 1 k  2 k


2 ( k )

2
k 2 k  1



.712319821093014476375... 
 (1)k H k  (k )  1 
k 2
3
.71238898038468985769... 
4 
2
4 .71238898038468985769... 
3

2
.712414374216044353028... 
log 7 4
.71250706440044539642... 
(1) k

3
k  0 2k  1

1

k
  k  k  1 log
k 2
( k  12 )!( k  12 )!

k 1 ( k  1)!( k  1)!

2k (k  1)


 2k 
k 0
 
k 


sin 3 2 x
0 x3
k  1

k 
.712688574959647755609... 




B2 k 2 k
 1  
coth  1  
2
2
e 1 2
k  0 ( 2 k )!

J948


 
=
1 .71268857495964775561... 
sin x / 2
dx
x
0 e 1


coth


2
 1 
J948
4k  1
2
3 log 3
 1  2
 l    l  
3 .7128321092365460381...  3 log 3 
2
 3  3
.71296548717191086579...  cos1 si (1)  sin 1 ci (1)   sin 1 
2
2
2
k 1
Berndt 8.17.8
1

 
=
  log x cos(1  x ) dx
0

1 .713009166242679...


1 
 dx 
( x) 
 1  
2
1
.71317412781265985501... 
.71342386534805756839... 
(1 / 2)

(5 / 6)(1 / 3)
 (1) k 
1 


3k 
k 1 

2
log 2 1

 
16
2
4
J1028
1
 x arctan
2
x dx
0

1 .71359871118296149878... 
 (3k  1) 
k 1


1 .714264112625455810924... 
1
 (k  1)(2k  3) log k
k 2
.714285714285714285
=
1 .714907565066229321317... 
5
7

2G

2
H
.715249551433006868051...   k k 2 
k 1 2 k

.715636602030672568737... 
 cos
 (  4)



1


1 3
arcsin x
.715884762286616949300... 
dx
log
 
2
2
3
0 x (1  2 x )
2

 1   k (k  1) 
k 2

1
.71616617919084682703... 
3
1
 2 
4 4e
1
1
cosh x
sinh x
0 e x dx  1  0 e x dx
.71637557202648803549... 
.71653131057378925043... 
2 2

e 1 / 3

sin



1 
2 

 1  2k
2
k 2
k
(1)

 
k
k  0 k!3

1
.716586630228190877618... 
.716814692820413523075... 
1 .716814692820413523075... 
log 5  arctan 2  2   log(1  4 x 2 ) dx
0
7  2
8  2 
(1) k


2
2
k 1 ( k  1 / 4)


256

125

log
.716890415241513593513... 
1
e2  1
log

2
2

.717740886799541002535... 

k 2
8 .7177978870813471045...

.718233512793083843006... 
2
Lk

k
k 1 4 k
.716863707177261351536... 
1
ex
0 e x  e x dx
 (k )  1
k k 2
76  2 19
2 3
3 3
CFG G1

.7182818284590452354...

 
1 .7182818284590452354...
2 .7182818284590452354...


1
1
1

 
 
k  2 k!
k  0 k!( k  3)
k  0 k!( k  1)( k  2)

1
= 

k  0 ( k  1)!  2 k !



k2
k
Hk

 e  1  
 
 
k  0 ( k  1)!
k 1 k!( k  2)
k  2 k ( k  2)!

1
 2i / 
 e  i
  
k  0 k!

e  2 
=


log x
dx
2
(

1
/
)
x
e
0

k3
3 .7182818284590452354...  e  1  
k  0 ( k  1)!
.718306293094622894468... 
=
 
i  
i 
 

csch
 log
   log 1 
1 

2
2 
2
 2



1 

k 1  ( 2k )
(
1
)
log1  2 




k
2 k
 2k 
k 1
k 1
.71837318344184047318... 
 2 log 2 log 3 log 2 3 1

12

=
e
x
0
.718402016690736563302... 
2

1
1
 Li2     Li2  2  
2  3
2
4
x
2

x3
0 (e x  1)2 dx
6 (3)   (4)  

.719238483455317260038... 

 (3k  1)

k
3
 (k )
.71967099502103768149...  
k  2 k!
k 1


 3k
k 1
k
3
1

11
log x
.71994831644766095048... 
 2
dx 
5
48
0 ( x  1)
.72032975998875729633... 

4

=
tanh


 4k

2
k 1
2



sin 5 x cos x
0 x5 dx
1
 4k  2
J950, GR 1.422.2
sin 2 x
 sinh 2 x dx 
GR 3.921.1
0


 
.7203448568537890207...
sin x
 e x
0

H ( 3) k
  k 
k 1 2 k
=
.720383533753944655666... 
2 .720416382101518554054... 
2 .720699046351326775891... 
.72072915625988586463... 
.721225488726779794821... 
.72123414189575657124... 
.72134752044448170368... 
e
1

cos
2
 5
cosh
2

5 (3)  2 (2)
 3

2

19
k 2
k

2


 1  k
k 1

2

1

2 
(k  1) 
(1) k

k  0 ( k  1)( k  1 / 3)
 19
2

4
3 (3)
5
1
 log 4 e
log 4
2
 3

tanh
arcsinh

8 .72146291064405601592... 
x
( k )


k
k 0
2
1
k 5
J153
.721631677641841900027... 
1 .72194555075093303475... 
.722066688901...

3 4 3
2


arccsch
7 7 7
3
3k
 2k 
 
k 
 (1)k 1
k 1
  log  


(1) k 

1


 k 2  k  1 
k 1 

13
.722222222222222222222 =

18
6 .72253360550709723390... 


1  x2
0 log(1  x) (1  x)4 dx
2e 3 / 2

2
k

1
 3
 

k 1  2  ( k  1)!
21 ( 3)  2 log 2
log( x ) log(1  x )
dx 


16
8
1  x2
0
1
.72256362741482793148... 
1 .72257092668332334308... 
.72305625265516683539.... 
3
18
9 log 3   3  3   (k )  1
  k 2 
2
3
k 2

4

3 3
.72319350127750277100... 
 
.72330131634698230862... 
 (k )  1
2
k  2  (k )

3
 k (3k  1)
k 2
sin x
 2  sin x dx
0

13 .72369128212511182729... 
1 .724009710793808932286... 
.72475312199827158952... 
1 .724756270009501831744... 
4 .7247659703314011696...

.724778459007076331818... 
.72482218262595674557... 

2e3  1

3

2
12


k 1
3 (3)

4
 (3  2 )
8
3k k
 (k  1)!

 (1)
k 1
k 1

k 1

k3
log x sin x
dx
x3
0
 
1
K 
 
16 3
, volume of the unit sphere in R7
105

2e
erfi
1

2
(1) k 1


k 1 ( 2k  1)!!

1
2


cot

2  3
3


k 1
(2 k )
3
k

 (1)k 1
k 1


 3k
k 1
2k
 2k 
k! 
k 
1
2
1
1 .72500956916792667392... 
.72520648300641149763... 

 2  log 2 2

6


40
log x
 ( x  2)( x  1) dx
GR 4.237.3
0
97
, probability that three points in the unit square form an
150
obtuse triangle
.725613396022265329004... 
1 .72569614761160133072... 
e  e log(e  1)

e 1
 log 3
2

Hk
e
k 1
k
 /2
 log(3 tan x) dx

GR 4.217.3
0


1
2 x dx
.72589193317292292137... 
tanh   sin 
2
2
 sinh x
0
 1
.726032415032057488141... 

8 3

1 
1 

.7264245534...


pk pk 1 
p k prime 
2
2
.72649594689438055318...  log 3  log 2
.726760455264837313849... 
.7268191868...

18 .726879886895150767705... 
2
4


  log(1  x 2 )
0
sinh
x
2 dx
x
cosh 2
2


1
1  2

2 
p ( p  1) 
p prime 

7 (2)  6 (3) 

 k  (k )   (k  1) 
3
k 2
8k  5k  4k  1
k 2 (k  1)3
k 2

sin 2k
i
.72714605086327924743... 
Li2 e  2i  Li2 e 2i   2
2
k
k 1
2 .727257300559364627988...   ( 2)   (4)
8
.72727272727272727272 =
11
 /2
1 sin 2
.72732435670642042385... 

  cos2 (cos x ) sin x dx
2
4
0


=

3
2
  
 
(1) k
 
.727377349295216469724...  e
k
k  0 k!
 3
.727586307716333389514...  Li2   
 5
1 / 

GR 4.373.5
2
7
 (3) 
2
2

1 

1 .72763245694004473929...   1 

k!2k 
k 1 

1 

.72784877051247490076...   1 

k !k! 
k 2 
sinh 
.72797094501786683705...  1 
.72760303948609931052... 
6 .72801174749956538037... 
.728102913225581885497... 
.728183137492543041149... 
.72856981255742977004... 


k
 ( k  1 / 2)
k 1
3


  
2

4 3

log 3

4

x2
1 x 4  x 2  1 dx

log 2 
2
 1 3
 2 Li3      (3) 
 2 2
12
1
=
1 .728647238998183618135... 
22 .72878790793390202184... 
log 2 x
0 ( x  1)3 ( x  2) dx
root of  ( x)  2
7 4

30

log 4 x
1 ( x  1)2 dx 

.7289104820416069624...

.72898504860058165211... 
.72908982783518197945... 
.729329433526774616212... 
21
dx
 4
64 2 0 ( x  1) 3
1


cosh 

45 (5)
log 4 x
 3
dx
64
x x
1
1  2e 2 


4 log 2
2 log 2  4   H k

3
3
k 1 k ( 2k  3)
1
cosh(cosh 2  sinh 2)  sinh(cosh 2  sinh 2)  e  
807 .729855042097228216666... 
2

1 e2
e k sinh k
=
e e  
2
k!
k 0
2
 1
1 .7299512698867919550... 
 cosh(log  )  cos(i log  )
2
.729439101865540537222... 


.73018105837655977384... 
 log 2
8
 /4
=
 2k   2 

  


k
2
k 0 



G


2
2 k 1
1
2 (2k  1) 2
2k
 /4
 x tan x dx   x cot x dx
0
Borwein-Devlin, p. 138
0
 /4

=
Berndt 9.32.4
 / 4  x tan x
cos 2 x
0
dx
GR 3.797.3
 /4
=
 log(cos x  sin x) dx

GR 4.225.1
0
 /4
=
 log(

tan x  cot x ) dx
GR 4.228.7
0
1
=
arctan x
 x(1  x ) dx
GR 4.531.7
2
0

1 .730234433703700193420... 
.730499243103159179079... 
.731058578630004879251... 
1
e1 / 4  
ex
1  erf    x 2 dx
2 
2
0 e
 1/ 3 


 2 / 3

e
1  tanh 1 / 2

  (1) k e  k
e 1
2
k 0
 (k )
 1  2  k!

=
J944
k 1
k 2

.73108180748810063843... 
2 2
log x
   3 dx 
27
0 x 1

x log x
1 x
0 x3  1 dx   0 1  x3 log x dx 
1

=
x(1  e  x )
0 e x (1  e3 x ) dx

1 .73137330972753180577... 
1 .73164699470459858182... 
.731771876673200892827... 
1
1  2
k 2
k
3


 1
e    Ei    log 2  
 2



.731796870029137225611... 
 3k  (3k )  1) 
k 1

2 .731901246276940125002... 
GR 3.411.26
 (k )

k
k 2
1/ k
e

7
k 1 k
 (k  7)!  
k 7

3

Hk
 k!2
k 1
3
(1  k  3 ) 2
k
( 1) k  2 k 
 

k
k  0 2 ( k  1)  k 



1  2k 
k
3  tan   k     k
1 .732050807568877293527... 
3
k 0 6  k 
k 0 6
5
3 .732050807568877293527...  2  3  tan
12

2
4
2
7
log 4 x
21 .732261539348840427114... 

 
dx
4
3
45
0 (1  x )
1

3 1 
.732050807568877293527... 
.732454714600633473583... 
1 .732454714600633473583... 
1 .732560378041069813992... 
 2k 
 
k 
AS 4.3.36

1

3 log 2
1
 1  i  Li2 1  i  

 G    log 2  iLi2 
i
2
8
2
 2 


 x 2  1  dx
1 log x  1  x 2  1

 (k )
.732869201123059587436...   2
k 2 k  1
=
.733333333333333333333 =
11

15
(2k  1)!!


k 1 ( 2k )! ( k  3)

 
=
GR 4.298.14
 2k 

k 1


.733358251205666769682... 
3 .73345333387461085916... 
 k  (3k  1)  1   k
k 2


 csc1   1 

.73402021044145968963... 

2
k
1
(1  k  3 ) 2
1 1
, 
 
1
2 
k 1


k 1
1
  k  4 (k  3)
k
0
(k )

2 2
( k  12 )!
.7340226499671578742...  16  16 log 2 

 2 ( 3)   3
1
3
k  1 k ( k  2 )!


 (k  3)
2
 
= 

(1) k 1

4
3
2k 1
k 1 2k  k
k 1

( k  12 )!
.73417442372548447512... 
2 1   
k
k 1 k ! 2

.7343469699579425786...


e cos1

2
e
 log x coslog x dx 
1
.73462758513149342362... 
2

2

sin
sinh
2


1 

  1  4  
4k 
2
k 1 
2
1 .734956638731129275406... 
sinh  2 csc h  
3
4
.735105193895722732682... 
 3


 1  k
k 1
2
1


 2k  2 
CFG G1

k3
1 .735143209673105397603...  3000  1103e  
k 1 k!( k  5)

( k )
2 .73526160397004470158...  
k
k 1 2
k 1

2
1  csch1   (2 1) 2 
.735689548825915723523... 
2
k 1 k  1 / 
3
2

2
k k k
.73575888234288464319... 

 (2,1)   (1)
e
(k  1)!
k 1

=
 xe
 1 x 2
(1) k 1 k 5

k!
k 1

dx
0

3 .7360043360892608938...

2

1/ 4

1 
4  dx

 log1  2 x
0
1 .736214725290954627894... 
1 i
1 i
      
 2 2
 2 2

1 .73623672732835559651... 
 H  (k  1)  1) 
2
k 1
.73639985871871507791... 
2 3 1 
(2k )!!
 

27
3 k 1 (2k  1)!!4k
 /2

 
=

0

.73655009813423404267... 
k
cos x
dx
( 2  cos x ) 2
 (1)
k
 ( k ) ( k  1)  1
k 2
k


.736576852723235053198... 

1
( x)
dx
xx

.736651970465936181741... 

k 1
.736842105263157894
=
14
19
 ( 2k )  1

e1 / k
  2
(k  1)!
k 2 k
2


1
  2k 
k 1
 
k 
CFG F17
1 .736901061414085912424... 
.73710586317587034347... 
 3 (3)
 ( 2 ) (i )   ( 2 ) (  i )

.73726611959445520569... 

( k )  1
k 2

2k  3

.73735096638620963334... 
k
3/ 2
1

k
arctanh
k 2
1
 F k (k  1)
k 1
3 .73795615464616759679... 


k
4 2
16


8 2


cosh  x / 4
  log(1  x ) sinh

 x/4

log 2  1 
2
2
0
GR 4.373.6

1 .7380168094946939249...

3
k
2



 2 (k )
3k
1
k 1
 2
.73806064483085791066...  Li3  
 3

p
4 .73863870268621999272...   k
k 1 k !
5
5 log 3 5 log 2
.73873854352439301664...  1 



4
3
4 3

1
3
(1) k 1 (k  1)
.73879844144149771531... 
 log  
3
2
2k k
k 1
512
 1
.7388167388167388167
=
   6, 
693
 2
k 1
.739085133215160641655... 
.7391337000907798849...

19 .739208802178717237669... 
k
root of arccos x  x 
 2 1
12
.740267076581850782580... 
7 .740444313946792661639... 

i
Li3 (ei )  Li3 (e  i )  

2
(1) k 1 sin k
Davis 3.34

k3
k 1

2 2

.739947943495465512256... 
(1) k 5k  (k )

6k
k 2

1


k
k 1 k  1
 3
6
(1) j 1

 k jk 
k  2 j 1


coth  3 
1

6

k
k 1
2
1
3
2 I 0 (2)  2 I1 (2)  I 0 (2)  3I1 (2)  I 2 (2) 
J124

k4

k 1 k!k!

.740480489693061041169... 
CFG D10
3 2

e
Fk


.740520017895247015105... 

2
k
e  e 1
k 1 e
7 (3)
5 2
.740579177528952263276... 
 log 2 

4
24

18k 2  11k  2
= 
2
3
k 1 2k ( 2 k  1)

1
.740726784269006256431...  
K  2 S1( k , 2)
.740740740740740740740 =
.7410187508850556118...

k2

k
k 1 4

3 log 1  3
 
=
2
5

sinh
2
5e
k 1
 (k  1)
k
3
 1
  hg   
 3
(1) k 1 Fk


k!
k 1

(1) k 1 k 2

k!
k 1

sin x
dx
x
1
2
2
  2 coth
3
7 5 log 2


8
4
3 .741657386773941385584... 


2
k
 2 (k
k
k 2


k
k 1
2
1
(k  1 / 2)
2
3
2
 1)
14
 (1) e

.741982753502604212262... 


e
0
1 .741433975699931636772... 
k 1
1

2
k
log(1  x )
dx 
x2
0
1
.741227065224583887196... 

 3k
3

.741027921523577355842... 
.741089701006868123721... 
k 3
k 2 (k )

2k

3 log 3  3


2
6
1

 (1)
k 1

20

27
1 .740802061377891359869... 

k
( k ) 1
 1
k 2

11 .743029813183056451016... 
 k  (k )   (k  1)  2
2
k 2
.7431381432026369649...

2G 
 log 2
2
1

arcsin x
 x(1  x) dx 
0
1

 
=
arccos x
dx 

1
x
0

GR 4.521.2
 /2

 
=
x sin x
 1  cos x dx 
GR 3.791.12
0
log(1  x )
1

 
=

1  x2
pf ( k )

e
0

.74314714161123874...


k 0


.74343322147602010957... 
dx 
GR 4.292.1
1
 (k  1)(k  1) log k
k 2
7 .743684425536057169182... 
.74391718786976797494... 
4
90
 11  6   6 (4) 

 (k )
k 1
k
 (2)

.744033888759488360480... 
k
 2 (2
k
k 1
k
 1)
log  (2k ) 

x 3 log x
0 e x  1 dx

 (k )
k 1
k2


1
k



k
k
k
1
k 1
k 1 2  2  2

k
=  k k
k 1 2 ( 2  1)
G
.744074106070984777872...   (3) 
2
3

3
sin k / 5
.744150640327195684211... 
 
k3
125
k 1
2

 1 (k )
k 1
2k

 /4
.74430307976049287481... 

cos x dx
0
.74432715277120323111... 
2 .744396466297114626366... 
2
8
1

arcsinh 
5 5 5
2

(1) k 1

k 1  2 k  1


 k 

log 


5 .74456264653802865985... 
1 .74471604990971988354... =
1 .7449110729333335623...

33
2


log x dx
2
1
0
 2x
Titchmarsh 1.6.2


2 .744033888759488360480... 

4 2
1
5e
1
(3 cosh1 2 sinh 1) 


4
8
8e
GR 1.443.5

=
k2

k 1 ( 2k  1)!

.74521882569020979229... 
  (k ) log (k )
k 2

5
( 1) k  2 k 
.7453559924999298988... 
  k  
k
3
k 0 5
.74562414166555788889...  sin sin 1
1 .745804395794629191549... 
16( G  log 2 )  24 
(1) k 1

2
2
k 1 k (k  1 / 4)


.74590348439289584727... 
  (k )  (2k )  1 
k 1
7 .7459666924148337704...
60  2 15

21 .74625462767236188288... 
8e
 (k )

2 .746393504673284119151... 
 (k  6)!

k 6


9 .7467943448089639068...

.7468241328114270254...

e 1 / k


6
k 1 k

95

erf 1 
2
( 1) k

k  0 k !( 2 k  1)

J973
1
=
e
x2
dx
0
5 .74698546753472378545... 
2 2
 1
 2 log 2 2  4 Li2    
 2
3
 ( k ) ( k )  1
 (k )
k 2
1
log 2 x
0 ( x  12) 2 dx 

.747119662029117550247... 
2 .747238274932304333057... 

 




1
2  5  15  6 5 
2
5  5  5  5  5  5  5  ... [Ramanujan] Berndt Ch. 22
=
59535 (7)  62 6
x log 7 x
 

24
( x  1) 3
0
1
17 .7476761359958500014...

  (k ) (k  2)

 1
 (k  1)
k 2 


.747727141203212136438... 
 
2 .747896782531657045164...
3G
9 .748110948000252951799... 
e 2   log 2  Ei (2)  

2k H k


k!
k 1

1 .748493952693942358428... 
.748540032992591012012... 

 
=
=
.74867010071672042744... 
2 .748893571891069083655... 
.748989900377804858183... 

H ( 3) k
11 (5)  2 (3)

  2
2
3
k 1 k

1
  2  coth    (2  i)   (2  i )  
2
2
1 i 
 
 (1  i )  i (1  i ) 
 2 


k 1

(1) k 1  (2k )   (2k  1)  2 


3
k 2 k  k
k 1
2

(1) k 1 ( 3k  1)( k  1)


 8G  
2
4k
k 1
7
8


1
1
dx
 (4)    4   4 
x
3
k 1 k
1
3 3
 
3
 / 2
.749306001288449023606...  e
.74912714295902753882... 

.749502656901677316351... 
1
 2  (k  1) 
k 1
k
( 1) k 1 ( k  12 )! 2 k

k!
k 1

.750000000000000000000 =
=
=
=
=

3
k
  i sin (i log 2)   k
4
k 1 3


 (2k )  1   21

k 1
k 2 k  1

1

2
k 1 k  2k


1

k  (2k )   (2k  2) 


2
k  3 k  2k
k 1

1 

1 


( k  3) 2 
k 1 

=
log 4 x dx
1 x3
7
4
11
2 .750000000000000000000 =

4
1 .750000000000000000000 =

p prime
1  p 2  p 4  p 6  p 8

1  p  2  p  6  p 8

372 .750000000000000000000 =
.75000426807018511438... 
1 .750021380536541733573... 
k6

k
k 1 3
   (1  e)  hg (e) 
cos
1
4
1

arctan 3 tan   1 

2
3
 2  cos d
0

e
 k ( k  e)
k 1
1
2k
k 0

log 4 k
( iv )
  (3)  
k3
k 2
1 .75013834593848950211... 
.7507511041240729095...
2

sin 2

.751125544464942482859... 
 1 / 4
.75128556447474642837... 
5 (3)

8
2
k
tan

H ( 2) k


k
k 1 2 k

 (1)
k 1
k 1
Hk

k2
log(1  x ) log(1  x )
dx 
x
0
1

 
=
.751300296089510340494... 
7 .75156917007495504387... 


3

3
4
1 

2  1   log1 
 dx
3x  1

0

(1) k 1

3
k 1 ( k  1 / 2)



2


t
 log  tan 4  dt 
2
0
J369, J605
K133

 


Borwein & Borwein, Proc. AMS 123, 4 (1995) 1191-1198

1
1
2k
k
.75159571455123294364...   
 log 2  2  Ei ( 2)   ( 1)
2
2e
k ! k ( k  1)
k 1
1
1
.75173418271380822855... 
I 0 2(1)1 / 4  J 0 2(1)1 / 4   0 F1 1, i   0 F1 1,  i  
2
2
 
1 .751861371518082937122... 

 (k  4)
k 0
k! 2 k

2
1 .751938393884108661204 

 2
1 .75217601958756461842 





e1 / 2 k

4
k 1 k

2
 

k 1   
k

1  4 3 3
1   1
 1 
( 2)  1 

 

3
,









 3,   

36  27
 3   108   6 
 3 
1  i 3 
1  1 i 3 
  Li3 

  Li3 
 2 
2
3   2 


 1  i 3 
i 3    1  i 3 
  Li3 

Li3 




3  
2
2



1
(1  x) log 2 x
= 
1  x6
0
4
.752252778063675049264
3 
=
 (3)

23
log x
.75267323992255463004... 
 2
dx 
4
96
0 ( x  1)
.753030298866585425452...  4  3 ( 4)
11 .753304951941822444427... 
e 2 I1 (2) 

1 .75331496320289736814 
1. 75338765437709039572... 
.75375485838430604734... 
2
 
=
cos x sin 4 x
0 x 4 dx
 2k  2  1


k  (k  1)!
k 0 


 2k 
k
  k  (k  1)!
k 0


1 

k
2
 1
e I0   
 2

GR 4.261.8

 
 1
e I0    1 
 2
(2 k  1)!!
k  1 ( 2 k )! k !


1
e
cos2 x
dx
0
coth  2   
2( k  1)


3
k 1 k  2 k



 1  2k
k 1
.75338765437709039572... 
(1) k

k  0 ( 2k )!( 2k )!


 ( 1)
k 1





1
1   1 i 2   1 i 2 
2
k 1
2 k  (2 k )   (2 k  1) 

.
75405876648941912567... 
 (1)
k
k 2
9 .75426251387257056568... 
.754610025770972168663... 
 ( k  2)
 (k )
15
 (5)  ( 2 ) ( 3) 
2



H 0 (1)  Y0 (1)    x
2
e
.75464783578494730955... 
4 .75488750216346854436... 
e 1
1 e 
e
e
 xe
x

dx
1 x
0
.754637885420670280961... 
3
Hk
(conj.)

2
k 1 ( k  1)

2
e
x
18 MI 4, p. 15
arcsinh x dx
0
dx
0

1
log 2


6
3
x
1

log 1  6  dx

x 
2
0
3 log 2 3
( 1) k

1
k  0 ( k  1)( k  2 )
 5  1
 5  1
2
  Li2 

 log 2 
.75539561953174146939... 

 2 
10
 2 


3 5 
2

 Li2 
=

30
 2 


5
 (5k )
1 .75571200920378862263...  
  e1 / k  1
k!
k 1
k 1
1 .75529829246990261963... 
  2 log 2 


3 .756426333323976072885... 

csc


2
cot

 2 Ei ( log 2) 
MIS


4
2 2
2
2
2

4
sinh 

6 .756774401573431365862... 
  1  4 
2
2
k 
k 1 

Bk
.756943617774572695997...  
k  0 ( 2k )!
.757342086122175953454... 
Berndt Ch. 9
csc

2

(1) k 1 k 2

2
2
k 1 ( k  1 / 2)

dx
 2 ( x  1)
x
0
.75735931288071485359... 
.7574464462934689241...

53 2
2(5)  (2)(3)  (3) 

.75748739771880741669... 
e
1
1
4
.757612458715149840955...   (3)  
9
k 0
2
k
( 4 )
2

k 2 Hk

4
k 1 ( k  1)

.757672098556372684654... 
2

8
 (3)


 k  (2k )   (2k  1) 
1

8
2
2
k2 1

3
2
k  2 ( k  1) ( k  1)
k 1


 
=
1 .757795988957853130277... 


 
=
=
.757872156141312106043... 
3
3 (3)

4

32
3 (3) 1
 (2  2i ) Li3 (i )  (2  2i ) Li3 (i )  
4
4
1
1
1 x
log 2 x
2
log
x
dx

0 1  x 4
0 1  x  x 2  x3 dx
e  1  erf 1 

e
x
0
1 .758268564325381820406... 

 
=
.758546992994776145344... 
.7587833022804503907...

dx
1 x
 


 


 3 3 cot

   2 3 cot
 9  csc 2
72 
3
3
3 



k2
 (2k )

k
k 1 3


 1



(1) k
k 0
k
1
 Ei (1)  Ei (2)  e log 2 
e
1
e
x
log(1  x ) dx
0
3  3 3 log 3
 2


 hg  
2
6
2
 3
.75917973947096213433...  2 ( 3)   ( 2 )
.758981249114944388204... 
1 .75917973947096213433... 
2 (3)   (2)  1 

 k  (k  2)  1 
2
k 1


 
=
k
k 1

.7596695583288265870...

e
1
x

.759835685651592547331... 
31 / 4
2
12
Hk
(k  1)
x dx
 e x
.759747105885591946298... 
.759967033424113218236... 
2

1
1
  Li2  ei / 2   Li2  e  i / 2  
16
2

k
(1) k 1
cos
2
2
k
k 1

 
1
( k )
.75997605244503147746...  e  2  
  j 
k  2 k!
k 1 j 1 ( k )!

FF
44
1 .760000000000000000000 =
  k kk 3
25 k 1 4

1
( 1) k
.7602445970756301513...  cos


2 k  0 ( 2 k )! 2 k


 
=


AS 4.3.66GR 1.411.3
1
k 

k 1

S1(2k , k )
.760339706025444015735...  
(2k )!
k 1
.7603327958712324201...


 1  5

 (1) k (1) k 
1

log(2  3 )  1   

6k  1 
3
k 1  6k  1
.760345996300946347531... 
(1) k (2k )!!


k
k 1 2 ( 2k  1)!!

=

=
 (1)
k
k 0

=
x
2
1 .76040595997292398625... 
17 .760473930229652590715... 

32 G

.761310204001103486389... 

5I 0 (2)  4 I1 (2) 
J86
( 2 k  1)!2

k
4
3
k  0 k ! 16 ( k  1)


k5
 (k!)
k 1
2

 (1)  (k ) (k  1)  1   k
k
 cot
1
k
k 1
1
J267
dx
3
16 log 2 

.7612801797083721299...
k 1
2k
 2k 
  k
k 
2
k 2
.76122287570839003256... 
 (1)
k 1
k!
(2k  1)!!

.761130385915375187862... 

J83
3
k 2

2
log
 1 (k )
2
(k  1)
k 1
k
 5  i 3 
3i 3
 3  i 3  
1    3  i 3 
  
  
  5
 
i  
 4 
 4 


4
2 3    4 






(1) k
=  2
k 0 k  k  1

u( k )
1 .761365221094699778983...  
k 1 k !

 (k  1)  1
.761378697273284672250...  

k!
k 1

e( k )
.761390022682049161058...  
k  2 k!



.761393810948301507456... 
e1 / k  1


k
k 2


2
x
 e sinh x dx
(e  1) 
2
4
.761549782880894417818...  log (  1)
0
e2  1
.76159415595576488812...  tanh 1  2
  i tan i
e 1
J148

=
1  2 (1) k e  2 k
k 1
4k (4k  1) B2 k

(2k )!
k 1

=
1
=
dx
 cosh
0
1 .761634557784733194248... 
=
.761747999761543071113... 
AS 4.5.64
2
x
97 6  4
2

 2 (3)  2 2 (3) 
(1   (3))  3 (5)
22680 72
6

H k H k 1

k4
k 1
 21 / 3
3 3


x
0
dx
2
3
3
2

e
1
(1) k 1
erf
.761964863942319850842...  1 
 e  
2
k!!
2
k 1
3
1 ( 3)  1  4
26
(1)  1 
.7619973727342293746...     
  
  (3) 
 3 54
 3 9 3
3
.761759981416289230432... 
.762054692886954765011... 
sin
e 1
K0   
2
 2

e
0
dx
x
2
1  x2


k2

4
k  1 ( k  1 / 3)
2 .76207190622892413594... 
 2 log 2
3
log 3 2
 1

 Li3   
 2
6
e 2  e 2

2
log(1  2 x ) log x
dx
x
0
1


4k


k  0 ( 2 k )!



16

=  1  2
2 
 (2k  1) 
k 0 

583
k7
2 .762557614159885046570... 
  (1) k 1 k
k!2
128 e
k 1

 (k )  (k )
.76258964475273501375...    0 k

2
k 2
3 .7621956910836314596...

cosh 2 
AS 4.5.3, GR 1.411.2
J1079
2 log 1  2   2arcsinh1  

1 .762747174039086050465... 

dx
1 e
(k  1)(k  2)(k  3) (k )
( 3)  3 
19 .763312534850599760254...      
4k  4
 4  k 4
x

0
arcsinh x
dx
x2
1


.763459046974903889864... 
2
3 e

4

k3

k
k 1 ( k  1)!2
(1) k
.76354658135207244811...  2  sin 1  cos1  1  
k  0 ( 2 k )!( k  1)
 1  3
2
5 .76374356163660007721...  6  log 2  7 log 2  l     l   
 4  4


.76378480769468103146... 
2
k 1
k
Berndt 8.17.8
Hk
H k 1
.76393202250021030359... 
3 5
.76403822629272751139... 
 (1)

k
 (k )

k
16
2
.76410186989382872062...    8G 

9
k 2

.764144651390776192862... 

1
 ( k  3 / 4)
k 1
2
  (k )  1
 (k )
k 2
  12

sin s k


4
6
k 1 k


1
( 1) k 1
  k

.764499780348444209191...   k
k 1 2  1
k 1 2  1
.764403182318295356985... 




=
1

k
k 1 2 ( 2  1)
1 
k
 ( 1)
k 1
( 4 k  1)
2 k ( 4 k  1)
2

Berndt 4.6
(1) k

k  2 S 2( k , 2)


 
=
6 .764520210694613696975... 
1 .764667736296682530356... 
5 4
 2 (2)


72
 (4)

 0 (k ) 2
k 1
k2

 (2)   (3)   (4)
13
17
sin  31 / 4 sinh  31 / 4
.7649865348150167057...  

2 2 3
.7647058823529411
Titchmarsh 1.2.10
=


2
(1) k

2 k
k  0 ( k!) 4

.76519768655796655145... 
J 0 (1)  I 0 (i )  
.765587078525921485814... 
2 3

81
(1) k

2
k  0 (( 2k )!!)

sin 2 k / 3
k3
k 1




.76586968364661172132... 
3
4 

 1  k
GR 1.443.5

1
 1  k (k  1)(k  2) 


8
13 (4)
31 2 3
 360 (5) 
 180 (3) 
 
63
2
6
4
k 1
6
4 .7659502374326661364...

 
.7662384356489860248...

=
 (1) k  (k )  1

3
5
log 
2
3
5
k

e
0
x
dx
2/3
e 1
.766245168853747101523...  1    Ei (1)  Ei ( 2)  
2 e
.766652850345066212403... 

 
=
8 .767328087571963189563... 
ex
0 (1  x)2 dx
1   3 coth  csch 2  i  (3,1  i )   (3,1  i  

 1
1 

i  

3
(k  i )3 
k 1  ( k  i )

=
1

1
x 2 sin x
x 2 sin x
dx


dx
0 e x  1
2 0 e x e x  1

2 1    Ei (1)   1 


kH k
 (k  1)!
k 1

.767420291249223853968... 
 (log 2)
2k

k 1
1 .767766952966368811002... 
(2k  1)!!k 2


k
2 2
k 1 ( 2k )!!2
5

 2k  k 2
  k 

k 0  k  8

1 .76804762350015971349... 
8 3
2i

Li3  (1)1 / 3  Li3 (1) 2 / 3 
81 3
3
 



1
=

log 2 x dx
log 2 x dx
0 x 2  x  1   0 x3  1 
GR 4.261.3

1
2 

csc h sinh  3   1  2

k  1
3
k 1 
21 (3)
 1  3i  3
.76816262143574939203...  6G  

   6 Li3 (i )  6 Li3 ( i ) 
 8 
2
5 .7681401910013092414...

 

=
 
.7684029568860641506...
 

8
arcsin x
= 
dx
x2
0
1

3
6G 
21 (3)

2
3

13

( k 2 )


  (2)  coth   
8
4
k2
k 2


1
=   ( 2 k )  1   6
2
k 1
k 2 k  k
1 .76842744993939215689... 
2
2
/ 12




2
( 1) k  1 / k
k 1
3 1  1
.76844644669358040493... 
  
6  22 / 3  3 

1 

4 .7684620580627434483... =  1 k 1 
2 
k 1 
3
.768488694016815547546...  G 


 .7687004249195908632... =

e
 x3
cos x 3 dx
0
1

 (k  2)!!  k !!
k 0
3 .768802042217039515528... 
cos 2( 1)
1/ 4
 cosh 2( 1) 
1/ 4


26k
 

k  0 ( 4 k )!


log( x 2  2)
2 .768916786048680717672...   arcsinh1   log 1  2  
dx
x2  1
0
6 .769191382058228939651... 
.769238901363972126578... 

 1 

2 
 
3
2
cos x
 4  2 cos x dx
0
 
cos log 2  Re 2i
1
1 .769461593584860406643... 
2   log 2  4 G  6    arccos2 x log x dx 
0

411
3
k 4Hk
31 .769476621622465729670... 
 15 log   k
61
2
k 1 3

2
1
.769837072045672480216... 
 coth    2csch 2   csch 2 sinh 2  csch 2 
4
8
4


1
k
=  ( 1) k  ( k )   ( 2k )  
  (1) k 1 k  (2k )  1
4 k 1
k 2
2

k
 
=  2
2
k 1 ( k  1)
7
.769934066848226436472...   ( 2)  
8

1
.76998662174450356193... 
2 J1 2   ( 1) k 1
k !( k  1)!2 k
k 0



 
.770151152934069858700... 
.770747041268399142066... 
.77101309425278560330... 
1
1 1  (1) k
cos
  
2
2 2 k  0 (2k )!

1
B
  kk
2( e  1)
k  0 k!2
2


1

1  4  
4 e 
4
J777
2
K1/ 2 (2) 
2

sin 2 2 x dx
0 1  x 2 
 /2
=
 sin
2
(2 tan x) dx
GR 3.716.9
0

.771079989624934599824... 

 2 ( 2k )
4k
1
2 2 log 2  35 (3) 
  G 
16
k 1
1 .7711691814070372309...

1 .771508654754528604291... 
.771540317407621889239... 

k
k 1
.772029054982133162950... 
.77213800480906783028... 
1 .772147608481020664137... 

k 1
k

 k
k 
3
4 (2)  4 (3)
k 
  

 (2k ,1) 
cosh 1
2

     e 1 
(2k ) 
2
k 1 k!
k 1 

1 .7720172747445211300...
   23k 
2
k
1

1

2 sinh 

2 2
e
 1/ 2
 (3) 
2
12
1
1 

 sin
 cos
 

2
2
log 2

cos x
 1 x
0
4
dx
1 .7724538509055160273...


 1
 2

1
( k  2 )!
= 
k 1 ( k  1)!
    

.77258872223978123767... 


 
=
1
 2 ( k  2)
k
k 0


 

 

 

( 2 k  1)!! k
 (2 k )! ( k  1)
4 log 2  2 
k 1



2

 (k )  1
2k  2
k 2
1

2 )( k  1)
k 0

dx
=  x x

0 e  e  1 / 2
=
 (k 
1 1
=
1
x y
  1  xy dx dy
0 0

1 .77258872223978123767... 
2 .77258872223978123767... 
=
=
H k 1
k
k 1 2

(k  1) H k

4 log 2  
2k
k 1

1

k  0 ( k  1)( k  1 / 2)

 (k )

k 2
k 2 2
4 log 2  1  
 2k  1
1
 k

k  4 (k  1)
k 0 

 

=
.773126317094363179778... 
sin 4 2 x
0 x3 dx
63

256

.773156669049795127864... 

k  2 p prime


 
=
2
1

0
x9 / 2
dx
1 x
1

pk
 (k )
GR 3.226.2


k 2
p
(k ) 

p prime
1

p( p  1)

k3  k
 (k )
= 
log  ( sk )
k
s  2 k 1
1
2 cos 2  2 sinh 2 
1 .773241214308580771497... 
4
3
.773485474588165713971...   (3) 
7
.773705614469083173741...  log 6 4
k 2
 

  (22kk)!

k 1
k
2

1 .773877583285132343802... 
1
F F
k 1
.773942685266708278258... 
40 .774227426885678530404... 
k
k 1
sin(e  )
15e 

k4


k  0 k!
1 .77424549995894834427... 
3 3
 64 
2
5 .77436967862887418899... 
1
sinh  
2
GR 0.249
 ( 1) k 1
( 1) k 1 



3 
( k  1 / 4) 3 
k  1  ( k  1 / 4)


 sin x sinh x dx
0
( 1) k  2 k 
.7745966692414833770... 


k 
k
k 0 6

 2k  2  k

1 .774603255583817852727...  64  44 2   
2  8k
k 1 
3

5

(1) k

k
k 0 2  k

.774787660168805549421... 
2 .77489768853690375126... 

8 .7749643873921220604...
2
 5
  (1)   
 6
2
(3k  1)( k  1)
 ( k  2) 

6k
k 1

77

(1) k 1
.776109220858764331948...  1  J 0 ( 2)  
k 1 k!k!

k 1
.776194758601534772662...   k 1 
k 2 k


3 .776373136163078927203... 
.77699006965153986787... 
  (3)
1
dx
0
 
 x
  2 x 

2 .77749955322549135074... 
.777504634112248276418... 
=
e1 / k


5
k 1 k


 (k )
 (k  5)!
k 2
1
1
 1  log(e  1)  Li2     Li2 (1  e)  
6
2
e

Bk
 1
Li2 1    
 e  k  0 (k  1)!
2
1

 
=
x dx
x
1
e
0
.77777777777777777777
=
7
9
17 .77777777777777777777
=
mil/degree
2 k 1
2e  
k  0 k!
1/ 4
10

14 .778112197861300454461... 
1 .778279410038922801225... 
2

pf ( k )
k 0 e k !

tan 1
( 1) k 1 2 2 k 1 (2 2 k  1) B2 k
.778703862327451115254... 
 

2
(2 k )!
k 1


1

k 1  ( 2k )
.778758579552758415042...   ( 1)
  1  cos 
k
(2k )!
k 1
k 1 
1 .77844719779253552618... 
.778800783071404868245... 
1 .778829983276235585707... 
1 .779272568061127146006... 
.77941262495788297845... 
1 .78005080846154620742... 
.780714435359267712856... 

e 1 / 4 
(1) k

k
k  0 k!4


2
log cos 2 x


0 1  x 2
1  e 2
 (3)  
 log

e2
2

arctan

e 3
3 3
3

6
cos


tanh
 5
2



k 1

GR 4.322.3
1
e
0
x
dx
 e x  1
1

 1  k (k  3) 


3

k
.7808004195655149546...   k
k 1 e  1

 (k )
.780833139853360038254...   k
k 1 2 k
1 .781072417990197985237... 

e 

k
 k!

k 0

=
=
lim
 1 ( n)
n log log n


1
1 p 1

n   log n
pn
lim
HW Thm. 323

1
.781212821300288716547... 
 Y1 (1)
.781302412896486296867... 
g2 



1  (1)  1 
1
1
 2 

     (1)      

2
2 
9
(3k  1) 
 3
 3   k 1  (3k  2)
J310
1
=
log x
dx
2
0 1 x  x

GR 4.233.1

.78149414807558060039... 
.781765584213411527598... 
6 .781793170552500031289... 
2k

2k
k 1 2
21
3 log 3

8
 

  
10
2
2 3
 3
 (3) 
3
1 
 k  (k )   (k  2)
2
4k  k 2  2k  1

k 3 (k  1) 2
k 2

=
2
3
 ( 1) k 1 

  1 
6k  5 
k 1 

1 .78179743628067860948... 
2
5/ 6
 2

.781838631839335317274... 
1 / 2k

1
k 1
1538 .782144009188396022791... 
113 .782299427444799775042... 
6 .78232998312526813906... 
1
 4
 ( 3)  

 2

1 2e
e  ee / 2 
2
46
k 1
k
sinh k
k!
1
1 .7824255962226047521...

2 HypPFQ [{1,1,1},{2, 2, 2},1] 
log 2 x dx
0 e x 

.78245210647762896657426...
 F  (2 k )  1
k 2
.78279749383106249726... 
k
1 8 log 2


6
9
(1) k 1
.7834305107121344070593... 

kk
k 1

.783878618887227378020... 
1 .783922996312878767846... 


 (k )  1
k 2
1
20

Fk

log( x  3)
dx
x3
1

1
 x dx
x
Prud. 2.3.18.1
0


5  1 e(1
5) / 2
5  3
5  2 5e
   (k F2)!k

5
k
k 2
4 .784000000000000000000 =
2 .784163998415853922642... 

 .7853981633974483096...

598

125


 (1)
k 1
k 1
Fk k 4
2k
3/ 2
2

1
  (1)  arcsin

4
2
(1) k

k  0 2k  1

AS 23.2.21, K135
=
 i arctanh i  Im log(1  i  Im Li1 (i )
=
1
1
4arctan   arctan  
2
3
Borwein-Devlin p. 73
=
1
 1 
4 arctan   arctan 

5
 239 
Borwein-Devlin p. 73
=
=
1
 1 
 1 
3 arctan   arctan    arctan 
 Borwein-Devlin p. 73
 20 
 1985 
4

1
GR 0.232.1
1  2
k 1 ( 4k  1)( 4 k  1)
( 1) k 1
J523
sin( 2 k  1)

2
k 1 ( 2 k  1)

sin(2k  1)
= 
GR 1.442.1
2k  1
k 1

sin k / 2
= 
k
k 1
3

sin k
= 
GR 3.828.3
k
k 1

(1) k cos( ( 2 k  1))
= 
    / 2 
Berndt Ch. 4
2k  1
k 0

2


[Ramanujan] Berndt Ch. 2, Eq. 7.3
=  arctan
2
 (2k  2) 
k 0

=

 

=
k 1

=
=
 1 
2

 arctan 2k
1
 k 1
k 1

 2
( 1) k 1 arctan 2 

k 
k 1
 arctan k
2
[Ramanujan] Berndt Ch. 2, Eq. 7.6
K Ex. 102, LY 6.8
[Ramanujan] Berndt Ch. 2



 
=

k 1
k ( k  1)


 
=
1
 1  (2k  1)
 (k  1 / 2)
k 1

=

 

GR 0.261, J1059

J1061

dx
dx
0 ( x 2  1)2  0 x 2  4 

=
2
2

dx
0 x 2  2 x  2 
x dx
1 x

dx
2
1
 4x
0
GR 2.145.4
4
0

=
1
=

dx
0 x3  x 2  x  1 
x
0
4
dx
 22 / 3
x dx

1  x4
0




log x
log x
=   2
dx 
2 dx   
( x  1)
( x 2  1) 3
0
0

 
=

e
0
 /2

 
=

0

=

0

=

0


 
=
 
=
=
sin x  x cos x
x3

0
 
=

0
0
4
 /2


sin x cos x
dx 
x2



2
sin x
dx 
x2
0
 
sin x
dx
2  sin 2 x
sin x cos x
dx 
x
 /2

dx
 e x

0

x
x dx
(sin x  cos x) 2
1  x cot x
dx
sin 2 x

=
 si( x) sin x dx
0


 
=
sin x
 1  e  x dx 
0
x
sin x cos 2 x
dx 
x


0
3
sin x cos x
dx 
x3


0
sin 2 x cos 2 x
dx
x2
 /2

 
 (sin
=
2
x) log tan x dx
0
1
=

0
x(1  x)
sin log x dx
log x
GR 4.429


 
=
 (1  x arccot x) dx
GR 4.533.1
0
1

 
=
   ( x) sin  x cos  x dx
GR 6/469/1
0
1 .7853981633974483096...



2k
1  

4
k  0  2 k  3


 k 

2 .785562097456829308562... 

k 1

.78569495838710218128... 

16
 sin


16

( 1) k 



1

8k  4 
k 1 

J1029
11
14
8
8
1
.78620041769482291712... 

arcsinh

9 27
2 2

( 1) k
 1 1
 
= 2 F1  11
, , ,   
 2 8
k 0  2k  k
 2
k
.7857142857142857142

cos
p (k )
, p(k) = number of partitions of k
k!
=

1 .78628364173958499949... 
k log k


2k
k 2
6 .78653338973407709094... 
 

4


1
log n k

 2k
n 1 ( n  1)!k  2

( 2k )  1
k 1

1 .78657645936592246346... 
1


k  1  ( k )1 ( k )

1 

p prime 

p
k 1
2k

1
k 1  
p 
Silverman constant

1 .78657645936592246346... 
1
  ( k ) ( k )
k 1
.78668061788579802349... 

=


 1   p
p prime
1

k 1
2k

1
k 1 
p 
1
 e  cosh(cosh 2  sinh 2)  sinh (cosh 2  sinh 2)  
2


2
1
e  e1 / e 
2

sinh k

k
k  0 k!e



dx
1
 5

5
x
k 1 k
1
k

2
( 1)
.78693868057473315279...  2 

e k 0 ( k  1)! 2 k
1

4
.786927755143369926331... 
 (5) 
.78704440142662875761... 
 (2)  2 Li3      (3) 
 1
 2

2 .787252017813457378711... 
e
k 1
3
1
k/2
1
e 4

9
4 .78749174278204599425...  log 5!
1 .787281880354185304548... 
1
3
2
log 2 x
0 ( x  1) 2 ( x  2) dx

Berndt 6.14.4

3k

k  0 ( k  2)!
1 
 log 2  2
.787522150360869746141... 
 coth 

csch  
2 4
2
4
2
1  1 i 
i
1 i 


  
  
    1     1 
4  2 
 2 
 2


k 1
k 1
=  ( 1)
k (k 2  1)
k 1
.78765858104243429...

.78778045617246654606... 
8  6 (3)  H ( 3)1 / 2

1
, lfac(k )  LCM (1, ..., k )

k  2 lfac ( k )
 

3 .78792362044502928224... 
i 

2  
3

( 2k )  1
k 1
15 (5)
  (5)   ( 4 )
22

1
i
i 
  log 2  i  Li 2     Li 2     
.78800770172316171118... 

4
 2   
  2
.78797060627038829197... 

1 .78820673932582791427... 

k 2
0
(k ) log  (k ) 

log 2  log sin 2 
1  x cos x
dx
x
0
log x
dx
2
4
x
1

 log (k )   log (n  k )
k 2
k 1
(1) B2 k 4
( 2 k )!( 2 k )
k 1

.788230216655105940660... 



n  3 k 1
2k
[Ramanujan] Berndt Ch. 5
2
=
.788337023734290587067... 


4
coth 
[Ramanujan] Berndt Ch. 5
 (3)


3
2k 3
.7885284515797971427... 

  k ( k  1) (2 k  1)  1   2
3
2
16
k 2
k  2 ( k  1)



 1  1 
 (k )  1
log k
    log1      2
.78853056591150896106...  

k 1
 k  k 2 k  k
k 2
k 2  k


 
=


 (k )  1
n 1 k  2

1 .78853056591150896106... 
k
n




1
1
  Li  k   k 
n 1 k  2

n

 k  1  (k )  1
k
k 2
4
5
8 .78889830934487753116...  8 log 3
1 .78885438199983175713... 
.789473684210526315
=
4 .78966619854680943305... =
15
19

.78969027051749582771... 
   k2!kk!
2 I1 2 2 
 ( 1)

k 1
k  (k )  1
log k

188
4
k 2H
 44 log   k k
81
3
k 1 4

1



k
k 1 2 H k



k!!
1
1





k 1 k !!( k  1)
k  0 ( k  2)!
k 1 ( k  2)!! k!!
3

Hk

3
k 1 ( k  1)
k
5  5
log 6  Li1     k
 6  k 1 6 k

(1) k 1


csc
1   2
2
2
k 1 k  1 / 2
i
 
2

coth     i    (i ) 2   (1) (1  i )   (1) (1  i ) 
2
2
4

Hk

2
k 1 k  1
k 2
2 .789802883501667684222...
.790701923562543475698...

 .7907069804229852813... 
.7916115315243421172...

1 .791759469228055000813...
1 .791831656602118007467...
1 .792363426894272110844...

 
=
log 3
.792481250360578090727... log 4 3 

2 log 2
7
2 .79259596281002874558... 
2



2
1
x
dx
 2 x

.792902785140609359763...
e
k 1
2
1 .793209546954886070955... 
.7939888543152131425...

.794023616383282894684... 

2
k
1
k 2
 
 (1) k 
1 


4k 
k 1 
(1 / 2)

(9 / 8)(3 / 8)

J1028
2 e 
Berndt 8.17.17




  xesin 1x dx
i
.79423354275931886558...   Im  (i ) 
 (1) (1  i )   (1) (1  i ) 
2
(1)
x
0

1 .79426954519229214617... 
  (k  1)  1 H
k 1
9 .79428970264856009459... 
25 .794350166618684018559... 
1
 2
 I 0 ( 2 e)  I 0    
 e
4
30 log 2  20 
.795371500563939690401... 
 1 / 5

.79569320156748087193... 

sinh k cosh k
( k !) 2
k 1

3
 (3)
.794415416798359282517... 
.7953764815472385856...
2k

k3


k
k  0 2 ( k  1)( k  2)
1
1
HypPFQ[{111
, , },{ ,2}, ] 
2
4
sin
( 1) k

 2k 
k 0
( k  1) 
k


2
 1
 1
.795749711559096019168...  Li2    Li3   
 e
 e
.7958135686991373424...  tan 1  tanh 1 
4 .7958315233127195416...

23

2
H
.795922775805343375904...   kk 
k 1 3
( 1) k 1
1  e x
.79659959929705313428...    Ei ( 1)  

dx 
k !k
x
k 1
0
= HypPFQ[{1, 1},{2 , 2}, 1] 

1
J289
0

 
=
 xe
xe x
dx

1
=
log x dx
ex
0

( 1)k 1

k 1 ( k  1)!  k !
4

k 1 ( 2 k  1)
 8    Ei( 1)   ( 1)

k !k
k 1
( 1)k


2
k  0 k !( k  1)

=
8 .7965995992970531343...
 

.796718392705881508606... 
3

Berndt 2.9.8

(3k )  1
k 1

.79690219140466371781... 
k 2
.797123679538449558285... 
.797267445945917809011... 
  (k )



   (k  2)  1

1

x
  ( x log
dx
1)( x  2)
log 2 2
  log 2 2 
9
2
0

1
2
(1) k
log  1   k 1
2
3
k
k 1 2

log 2 k

k  2 k ( k  1)
2
.79788456080286535588... 
2
1 .7977443276183680508...

.797943096840405714600... 


9 .7979589711327123928...

.7981472805626901809...

2   (3) 
k 1


3
k 2 k


 (1)  (k )   (k  2) 
k
k 2
96 

3
tanh

2 3
1 

3

2 3
3  3e 
3


 
=
  (6k  2)   (6k  3)   (6k  5)  (6k  6)
k 0
1 .7981472805626901809...


 

 
tanh




2 3

2 3
3 3  3e  3
 3  i 3
i 3   3  i 3
 
  

= 1 
3   2 
 2  
3


=
 1
1

k 0 k  k  1

2
GR 4.261.4

2  (3  5 )e 5
Fk
e 1 5  / 2
 
5 5
k 1 ( k  1)!

k 1 k ( k  1)
.798512132844508259187...   ( 1)
 (k ) 
k!
k 3

e2  1
2k
.798632012366331278403... 
 
8
k  0 k !( k  1)( k  3)
2   2  (1   ) (1  
  1   (2   )
.79876560170621656642... 



 (  1)
3 .79824572977119450443... 



 
=
.798775991447889498103... 

   (k )  1
1


k  2 k (k   )
k 2
k 2
3 cos 2

 sin 2 
2
2
(1) k 1 4k


k 1 ( 2k )!( k  1)


.7989752939540047561...

.79917431580735514981... 
1 .799317360448272406365... 
.799386105379510436347... 
2 sin 2 1

k 0

 (2 k  1)
k 1
2k k

( 1) k

 ( k  1)!( k 

1
2
)

1 
1 
   log 1  2 
 2k 
k 1 k

 
kj
( k ) k
3

  k j
2 k  2 2k
k  2 j 1 2

1
log 2



4 4 sinh 
2
1 
i
i

1 i 
1  i 
  1     1     
  
 
8  2
 2 
 2
 2 
(1) k

3
2
k 0 k  k  k  1

=
.79978323254211101592... 

4
1  e 
4
 /2

 cos(2 tan x )dx 
0
GR 3.716.10
4  (1) k

5 k 1 4 k

F2 k
=  k
k 1 4

(1) k 1 (2k )
.800182014766769431681...  

2 k 1
k 1 ( 2k  1)!2
.800000000000000000000 =
J352

1
1
 k sin 2k 
k 1
log( k  1)

k2
k 1

1 .8007550560052829915...




 
=

1
6 .80087349079688196478... 
.80137126877306285693... 
.801799890264644999725... 
12 .80182748008146961121... 
.8020569031595942854...

.80257251734960565445... 
1 .802685399488733367356... 
.80269608533157794645... 
H ( x)
dx
x2

 ( 3) ( k )
k 1
k

2 ( 3)

3
7 (3) 1
 
8
4
k 2 (2 k  1)

4k
k 1

log 9!
2

5
1
14

arccsch 2 
5 5 5
 ( 3) 

 (4 k )
k 1
k!


( 1) k 1

k 1  2 k 


 k  1

 14

  e k  1
3 3

sin

2
3


1 
2

 1  3k
k 2

.8027064884013474243...


si (2)

2
( 1) k 4 k

k  0 ( 2 k  1)!( 2 k  1)


3 ( 3)
( 1) k
1 .8030853547393914281... 

 2 ( 3)  2 
3
2
k  0 ( k  1)
1
log 2 x
dx 
 
= 
1 x
0
GR 4.261.11
1

 
=
x log 2 x
0 (1  x) 2 dx 


 
=

1
GR 4.261.19

log 2 x
dx 
x2  x
x2
0 e x  1 dx 
1 1

 
=
.80325103994524191678... 
log( xy )
dx dy
0 0 1  xy
 
4G  1 

  log 2 
2
2
4
 /2


0
x 2  sin 2 x
dx 
1  cos x

.8033475762076458819...
1
  k 1
2
k 1 2

1 .8037920349100624...

 (1)
k  1 1/ k
e
k 1
.80384757729336811942... 
6 3 3
3 3
 12
2 .80440660163403792825... 
2
log 5
.80471895621705018730... 
 Relog 2  i
2

1 

1 .804721402853249668...
   1  k 
2 k
k 1 

e1/ k
2 .80480804893839018764...   4
k 1 k
1 .8049507064891153143...
.8053154534982627922...


4

 (3k  1)
k 1
k!



.80612672304285226132... 
.8061330507707634892...


e

1
1/ k 3
k
k 1
 1 csch 2 csc h 2 ( (1  i ) 
1 2

  2 

  
4
2
2
4
4


 ( i ) 
2

Hk
 
= 
2
k  1 k ( k  1)

1

.80568772816216494092...    1  k 
6 
k 1 
 

 
 2 ( i )
4

 1
L1/ 2   
 2
2
4

9 3

k 1
k 1
 
k
k 1
1
k
  2k  
k


 (i )
2

 2 (i )
4


 

k ( k  2)
 4   5
=        

 3   3 k 1 ( k  1 / 3)( k  2 / 3)

=
 (x
dx

 1) 2
3
0
1 .8061330507707634892...
4
 1 

9 3


 
=
 (x


k 1
k 1
 
k
  2k  
dx
 1) 2
3
0
.8063709187317308602...
 1
 2  2 log 2  log 2 2  3 log 3  2 log 2 log 3  log 2 3  2 Li2   
 3
1

 
=
 log x log( x  2) dx
0

.80639561620732622518... 
dx
0 e x  x 

e
0
x dx
x
x
 2
9
  
5
 10 

k
1 .80649706588366077562...  
k  0 k ! 1
1 .806457594638064753844... 
.806700467600632558527... 

2 2

sin




1
  1  2(2k )
2



2 2
k 1
2
.80687748637583628319...  128  32 G  4   8  48 log 2 
log(1  x ) log x
dx
x 3/ 4
0
1

 
=
J1061


21/ k
2 .80699370501978937177...   2
k 1 k

 ( k 2  2)
.8070991138260185935...  
k2
k 2

( 1) k 
.80714942020070612009...    1 

2k 
k 2 
2 .80735492205760410744...  log 2 7

.807511182139671452858... 
 4 1  1 
      ,1 
 3 3  3 
.80768448317881541034... 
13
1
e
0
x3
dx

2 .80777024202851936522... 
dx
  ( x) Fransén-Robinson constant
0

.80816039364547495593... 

2 .8082276126383771416...
 (k )  1

k  2  (k )


4 (3)  2
=
(1  x ) log 2 x
0 1  x dx

4 (3)
=
log(1  x1/ 2 ) log x
dx
0
x
1
4 .8082276126383771416...
1
1 1


5 .8082276126383771416...
log( x 2 y 2 )
dx dy
1  xy
0 0
=
 

4 (3)  1 

 k  (k  1)   (k  2)  2
2
k 1
.80901699437494742410... 
.80907869621835775823... 
.8093149189514646786...

.80939659736629010958... 

 
=

2

1 5
4
2 log 2   
1
2

 ( k  1)
k 1
2k


1
2
  , 
  3
1
1
 
cosh

1/ 3
3
3  ( 1)   ( 1) 2 / 3 
 2 
1

 5  i 3  5  i 3
 


 2   2 
1
   (3k )  1

 
=   1  3   exp  

 k 1

k 
k
k 2 

Hk ( k  1)
1 .8098752090298617066...  

2k
k 1


1
.81019298730410784673... 
120  4 2  24 ( (4)   (5)   (6))    log(1  x) log 4 x dx 
0
7 .8102496759066543941...

.81040139440963627981... 
61 
1

 2 csch

2
2
( 1) k

2
k 0 2k  1

k

 4
11 .81044097243059865654...    
 
k 1 k
/2
4 .81047738096535165547...  e
 e i log i 
8
4
.81056946913870217...



2

3 ( 2 )
1 .81068745341566177051... 

(2 k )

2

 ( 2 k  1)
 (2 k  1)
2
k
k 1

 ( 1) k 1
( 1) k 1 

3  

2
( k  1 / 6) 2 
k  1  ( k  1 / 6)
36  2 2
k 1


1 
 sin 

  1 
 1
 k2 
k 2 
.8108622914243010423...

.810930216216328764...
 1

 2 (log 3  log 2)     ,1,1 
 2




 
=
e
x
0
.81114730330175217177... 

dx
1/ 2
 15
tanh

2
k6
551 .8112111771861827781...  203e  
k 0 k !

log( k / ( k  1))
.81139857387224308909...  
( k 2  3)
k 2
1 .81152627246085310702...  arccosh 
15
(1) k


k
k  0 2 ( k  1)


k
k 0
2
1
k 4

.8116126200701152567...

1
2

arcsinh 1 
2
4
( 1) k 1 (2 k )!!


(2 k  1)!!
k 1

( 1) k 1 4 k

 2k 
k 1
 
k


 
.8116826944033783883...
=


Hk

2
k
 2k
4 log 2  2 log 2 2  1 
k 2

Hk
2

1 .8116826944033783883...  4 log 2  2 log 2  
2
k 1 2k  k


 
=
log 2   H k 1  (2k )  1
k 2
1 .8117424252833536436...

MHS (2,2) 

1
2
k  j 1 k j

2
log1  k 2 
  Hk ( ( 2 k )  1)   
1  k 2
k 1
k 2

.8117977862339265120...
3 (4)

4

143 .81196393273824608939... 
.81201169941967615136... 
1 .81218788563936349024... 
.81251525878906250000... 
.8126984126984126984
=
.81320407336421530767... 
.81327840526189165652... 
1 .81337649239160349961... 

 5 (k )
k 1
k!

k 4
6
k 1 2 k
1

(

)

e2
k!
2e
3
1
1
1
1
 2  2 2  2 2  ...
2 2
2
22
256
 1
   5,  
315
 2

2
 2
1
sin
 1  2

3
3
9k  1
k 1

21 / 4  3   3 
 1
      3 / 4       sin
 4
 4  4

8
J1064


sinh 2
4k


1 .8134302039235093838... 
2
k  0 ( 2 k  1)!

4 

=  1  2 2 
 k 
k 1 
11
 ( 3)
.81344319810303241352... 
 6 log 2  4 log 2 2 
4
4
1
2
(1  x ) log(1  x )
= 
dx
x
0
GR 1.411.2
GR 1.431.2

4 log 768
kH

  k2 k
9
18
k 1 4

1
 1
 1
 2
         hg     hg    
1 .81379936423421785059... 
3
 2
 2
 3
 3
2
2
 
=

sin
3
3

d
 
= 
2  cos 2
0
.81354387406375956482... 




 
1 

=  log 1 
 dx 

x ( x  1) 
0
 /2

 
=
log(1  x 3 )
0 x 3 dx
sin x
 sin x 
E
 dx
2
x) / 4  2 
 1  (sin
0

GR 6.154
2

k2
 (k  1)  1
.814748604226469748138...  4  2 
 2 log 2  
6
k 1 k  2

2
sin 2k
2
.8149421467733263011... 
 
  3
3
3
k
k 1
.8152842096899305458...
3 2 2 3

 
3
GR 1.443.5

1  x 4
0 log 1  x  3 dx

47 .81557574770022707230... 
5 5
log 2 x dx
 2

32
x 1
0

=
5 5
x4


 24i Li5 (i )  Li5 (i )   x
16
e  e x



 
x 4 dx
= 

cosh x
0

.815690849684834190314... 
1
E
k 1
.81595464948157421514... 
.8160006632992495351...

.816048939098262981077... 
2 .816190603250374170303... 
3 .816262076667919561...

2 .81637833042278439185... 

GR 3.523.7
2k
3
38

2  1
 tanh(log  )
2  1
 3
   
 4
1
2k  1 

 1  2k  2 
2
 (3) 

12
k 0


19

24
k 1
1  2 
I    I 2 e
2  0 e  0
6
1
2 
12  
2   (2)

Hk Hk
 k (k  3)
k
   cosh
( k !)

2
k 0



k 1
f (k )
k2
Titchmarsh 1.2.15

.81638547489578...

  (3k  1)   (3k  1)   (3k )  1 
k 1

.8164215090218931437...


k 1

=
(
k 1
.81649658092772603273... 
 ( 2k )
2 1
1
k
2)
2

3
2k



 (2k  1)
22 k 1  1

log k
   k2  
2
k 2
k 1
( 1) k  2 k 


k 
k
k 0 8

1
2
k 0
2k

J168
.81659478386385079894... 
3 log 2

8

1 .81704584026913895975... 

k 1
1 .81712059283213965889... 
3
 1  x 2  dx
log
  1  x  1  x 
GR 4.298.12
H ( 3) k
k!
6

5 .8171817154095476464...

.817222646239917754960... 
k5
33  10e  
k  1 ( k  2 )!


e 2
 K1 / 2 (1) 

cos x  sin x
dx 
x ( x  1)
0



 
=
cos x  sin x
x dx 
x2  1
0

1
.81734306198444913971...   ( 6) 

5
.81745491353646342122... 
=
4 .81802909469872205712... 
.818181818181818181
=

1

6 
k 1 k
Prud. 2.5.29.23

dx
x
6
1

3
9 2
cos 3k
 
 

2
4
6
k2
k 1
1
Li2  e3i   Li2  e 3i 
2
e 

GR 1.443.2

9
11
1
arccsch 
3

1
4 .818486542041838256132...  
k 0 K k (1)
1 .81844645923206682348... 

1 .81859485365136339079... 
2 sin 2   ( 1) k 1
k 1
2 .81875242548180183413... 
.8187801401720233053...


4k
(2 k  1)!
3
11
   2 log 2  log   

log x
 cosh
0

 ( 3)
1 .81895849954020784402... 
 ( 2)
3 .819290222081750098649... 
2 6  32 log 2 

k3


k
k 1 2 (k  2)
2
x
dx 
GR 4.371.3
k 2  2k  2
26 .819615475040601592527...   3 sinh  csc  3   2
k 1 k  2 k  2


1 
1
log 2 x
1 .81963925367740924975... 
dx
   4,3,    2
4 
2 1 x  4


1
 (k )
  k 
.82025951154241682326...   k
k 1 e  1
k 1 e

 (8k  6)

1 .82030105482706493402... 
 ( 2k  1)!


k 1

1 .82101745149929239041... 

2
sinh
k 1
1
k4
 ( 2 k )
k 1

.8221473284524977102...
k
Berndt 6.14.4
  2
( k ) 1
k 2
 1

1 .82222607449402372567... 
3 ( 3)  2  ( 3)  2 ( 3) 

0

1 .82234431433954947433... 

 ( k  3)
k!
k 1
2
.82246703342411321824... 

12
x 2 log x
dx
ex  1
e1/ k  1
 
k3
k 1


(1) k 1


k2
k 1

Hk


k
k 1 2 k

S1(k ,1)


k!k
k 1

  ( k )   (1)

k
 1 


2
k 2 


 
=

 
=

 
=

 
=


 
x
dx 
= 
1  ex
0
log 2

 
=

0
 
=

1
log x
log(1  x 2 )
dx   
dx 
x2  x
x
0
1
x
dx 
1  e x
log(1  x )
0 x dx 
1


J306
GR 3.411.5

 log1  e  dx 
x
Andrews p.89, GR 4.291.1
0

 
log(1  x 2 )
=  
dx 
x
0

 
=
1
2
log x
 x  1 dx 
1
GR 4.295.11
[Ramanujan] Berndt Ch. 9
1

 
=
log x
dx 
x 1
0

GR 4.231.1
1

 
=
  log1  e x  dx 

0


 
=
log(1  x 2 )
0 x(1  x 2 ) dx 
GR 4.295.18


 
 1  x2  x
dx 
=  log

 x2  1  x2
0


 
=

GR 4.295.16
1  dx

x
 log1  x 
1


 
log( x 2  1)
dx 
= 
x ( x 2  1)
0


 
=
e
x3
1
x2
0


 
=

0
dx 
e 2 x
x
ee  1
dx 
GR 3.333.2


 
=

 
=
xe  x
0 sin x dx 

x2
0 cosh 2 x dx

1 .82303407111176773008... 
 ( 4 k  2)
 (2 k  1)!
k 1

.82329568993650930401... 
2
k 1
.8235294117647058
.8236806608528793896...
=
14

17


1 .82378130556207988599... 
1 .82389862825293059856... 
.82395921650108226855... 
2
18
2
3
(k )
2 k 1



 1
2

 sinh k
k 1
1
2
6

 2 
0
3

( 2)
17
3 log 3

4


sin 3 x

2
k 1 x
x log 2 xdx

ex
.82413881935225377438... 
1
  ci ( )  log   
2

 /2

0
sin 2 x
x


1
e
k
k
.82436063535006407342... 
 
k 1  
k  
2
k  0 k !2
k  0 k !2
k  0 (2 k )!!
2

2
.82437753892628282491...  5 
 2 (3)   k 2  (k  3)  1
3
k 1

k 1
=

2
3
k  2 k ( k  1)

1
16 

.82447370907780915443... 
sin 2 sinh 2    1  4 4  
 k 
4
k 1 

1 .824515157406924568142... 
=

5
3 5
EllipticTheta 2 2, 0,

4
2 


1
 2 k 1

5

  4k 2  1
k  0 F 2 k 1

4 
 3
 3
.82468811844839402431... 

 3 log 2  hg     (1) k 1    (k  1) 
3
2
 4
 4
k 1

3
 
= 
k 1 k ( 4k  3)

(2 k )
.8248015620896503745...   
k!
k 1

8 .8249778270762876239...  2

i  (1)  2  i 
k
(1)  2  i  
 
.825041972433790848534... 
    2
  
2
2
 2 
 2 
k 1 ( k  1 / 4)

(1) k k 5
1 .82512195293745377856...  J 3 ( 2 )  6 J 2 ( 2 )  7 J1 ( 2 )  J 0( 2 )  
2
k  0 ( k !)

2k  2
2
13 .82561975584874069795...  (1  e)  
LY 6.41
k!
k 0
9
9 .82569657333515491309... 
G

k
2 .82589021611626168568...  
k  2 k ! 1
k

(1) k p (k )


2k
k 0

.825951986065842738464... 

.82606351573817904...

 k k
k 2
4
k
3


1
  1  (1/ 2)
k 1

(1)
 k2  k 1
k




k (2 k  1)
3k
.82659663289696621390...  
 
2
2
k
3
k 2
k  1 ( 3k  1)

1  cos 2
( 1) k 1 2 4 k 1
2
.8268218104318059573...  sin 2 


2
( 2 k )!
k 1
 2  3
 log 2 2

  log 2 
3 .82691348635397815802... 

2
24
2

dx
2
 
=  log x sin 2 x

x
0


.8269933431326880743...


3 3
1 

  1  2  
2
9k 
k 1 
 cos 3  2
k 1
.8270569031595942854...
=
 0 


 1 / 3

 ( 3) 
.82789710131633621783... 
4 .82831373730230112380... 
k
3
8
1
1
 / 2 
1 e
1  ii
3 log 5
( 1) k  2 k 
.828427124746190097603...  2 2  2   k
 
k  0 4 ( k  1)  k 

1 .828427124746190097603... 
2 2 1 

(2 k  1)!!
 (2k )!!2
k 1
.82853544969022304438... 
1

log 2 3


0
k
x dx
3x
 1
5 .82857445396727733488...   12 Li4    
 2

x 3 dx
0 e x  1 / 2

.82864712767187850804... 
.8287966442343199956...

log k !

k!
k 2


1
 1
14 (3)  16    ( 2 )     2

1 3
 2
k  2 ( k  2)


 
x2
dx
=  x / 2 x
e  1
0 e
GR 4.424.1
GR 1.431


=
GR 1.412.1
16 .8287966442343199956...

14 (3)   

.8293650197022233205...


 (k )
2
k 1

.82945430337211926235... 

 1
  
 2
k 0
p prime
1 .83048772171245191927... 
1
2
)3
1
 k !log k !
k  21
.82979085694614562146... 
1
 (k 
1
2p

 2
k
(2)
 ( 3)   ( 2 )   


 ( k  1)

 ( k  1)
k 1
2
1  cos1 i ei  1 i 1  cos1  i sin 1
 i

sin 1
e 1
(cos i sin i  1)

.83067035427178011249... 
 log ( k ) 
k 2
5 .83095189484530047087... 
.83105865748950903...

2 .83106601246967972516... 
.8313214416001597873...

34
H (2)2/ 3 

3
log 7 
2
5
arccot

3
3
1



cot

2 2 e
e


k 2

log( x 2  8)
0 x 2  1 dx
 (2 k )
e
k


 ek
k 1
1

1
2

8 .8317608663278468548...
78

1

.83190737258070746868... 
 ( 3)


 (k )
k 1
k
3


  1 
p
1

p3 

.83193118835443803011... 
1 .83193118835443803011... 
2G  1 
x tanh x
dx
ex
0

2G  i Li2 (i )  Li2 (i )  



 
=
( k !) 2 4 k


2
k  0 ( 2 k )!( 2 k  1)
 /2
=

0


 
=
x
dx
sin x
x
 cosh x dx 
0
1

 
=
arcsin x
dx 
x2  1
0

Berndt 9
Adamchik (2)
1

 
=
arcsin h x
dx 
x2  1

0


 
=
arctan x
x
0
1  x2
dx 
GR 4.531.11
1
=
 K  (x ) dx
GR 6.141.1
0
1
=
 K( x
2
) dx
Adamchik (16)
0
.83205029433784368302... 
3 .83222717267891019358... 
3

13
( 1) k  2 k 


k 
k
k 0 9



1
9 log 3   3 
4

1
 ( k  1)( k  1 / 3)
k 0

.83240650875238373999... 
1 .83259571459404605577... 
.83261846163379315125... 
2
1
  3 log 2     e 2 x log x dx 
4
2
0
7
12
GR 4.383.1
 1/ 3
45 (5)
log 3 x
   log(1  x )
dx
x
8
0
1
5 .8327186226814558356...

.83274617727686715065... 
.8330405509046936713...

=

log 2
3

8 2

 (x
0
4
dx

 1) 2

k ( k  1)
 5  7
     

 4   4  k 1 ( k  1 / 4)( k  3 / 4)
 2
Li2  
 3
5
=
6

( 1) k
= 
k
k 0 5
.83327188647738995744... 
.83333333333333333333
 


 
=
1
 ( k  1)( k  2)( k  3) 
k 1
k5  k4  k3  k2  k 1

k7  k
k 2


 
=
J1061

 2k 
(2 k  1)!!
1





k
k 1 4 ( k  2)  k 
k  1 ( 2 k )! ( k  2)

 

=
|  ( k )|( 1) k  |  ( 2 k )| 2 k
.8333953224586338242...  

2k  1
4k  1
k 1
k 1

4k
k
.83373002513114904888...  cos1 cosh 1   ( 1)

( 4 k )!
k 0
1 .83377265168027139625...  root of  ( x )  x

 (2 k )

1 .83400113670662422217... 

1
1
 (2 k  1)!  k sinh k
k 1

k 1
15  90
( 2)  1

4
  90
( 4)  1
11
3
  2 e cos  
.83471946857721096222... 
3 e
2 
2
7 .8341682873053609132...

(1) k

k  0 (3k )!


pf ( k )
k!
k 0

1
sin 2 k
.83501418762228265843...   ( 3) 
Li3 ( e2i )  Li e 2i    3
k
4
k 1
2

1

.83504557817601534828... 

 8 log 2  8   3
2
3
k 1 k  k / 2

(1) k 1 ( k )
 
= 8

2k
k 3

( k  12 )!
 
=  2
1
k  1 k ( k  2 )!

h(k )
.835107636...
  2 
100 AMM 296 (Mar 1993)
k 1 k

1
.8352422189577681...
 
k  1 Fk k
3 .83482070063335874733... 




4 .8352755061892873386...

31 6  28350 (5)

84

1

0
x log 6 x dx

( x  1) 3
5

 (2  log 2)  2 G 
24
2
2
1 .835299163476728902487... 
135  
 3 2 ( 4 )
45

2
sin x  x cos x
.835542758210333500805... 
dx 
 
5/2
x
3
0


1
2  dx

 log(1  x) log 1  x
0
4
.83535353257772361697... 
Prud. 2.5.29.24



log 2
.83564884826472105334... 


3
3 3

( 1) k

= 

k  0 3k  1

=

1
2 .83580364656519516508... 

1 .83593799333382666716... 

1
J79, K135
K Ex. 113
dx

3
x 1

0
5 3
 27 
3 3
1
k 1
1
dx

2
x  x 1

  6k  5  6k  2  

e
x
0
dx
 e 2 x
 ( 1) k 1
( 1) k 1 



3 
( k  1 / 3) 3 
k  1  ( k  1 / 3)

1
 (k !!)!!
k 1

.83599833270096432297... 

 (k )
k
k 2

.8362895669572625675...


2
(1)


 1
1
  Li  k   k 
2
k 1
k
 k ! 1 
k 2
(1) k

k
k  0 ( k  1)!e


.8366854409273997774...

e  e11 / e  
6 .8367983046245809349...

2
.83772233983162066800... 
8

3 3

3k

k  0  2k  1


k


4  10

log x dx
/2
1
0
 x
4 .83780174825215167555... 
4 csc 2 
.837866940980208240895... 
Cosh int1
.837877066409345483561... 
log(2 )  1 
2

 (2 k )
 k (2 k  1) 
Wilton
k 1
1 .837877066409345483561... 
.83791182769499313441... 
.83800747784594108582... 
1 .838038955187488860348... 
1
cos

3
=
(1) k

k 
k  0 ( 2 k )! 3

AS 4.3.66, LY 6.110
3
37
sinh 
1


2
 ( 2  i ) ( 2  i )


Messenger Math. 52 (1922-1923) 90-93
log 2

1

2 

 1  k
k 2

 (4k  1)! 
k 0
k  2k  2
2
 2k  1
k 1

k
2
 4k
2 e2
.83856063842880436639...  log 2

J157
e 1

B 2k
=  k
[Ramanujan] Berndt Ch. 5
k 1 k!k

125
 (5k )
.83861630300787491228... 
  3
k
124 (3)
k 1
2
.83899296971642587682...  G

 k ( k  1)

95 .83932260689807153299...  e (  1)  

GR 1.212
k!
k 0

5
( 1) k 1 k 6
.83939720585721160798... 
1  
e
k  1 ( k  1)!
5
1 .83939720585721160798... 
   3,1
e
1
1
2

2 .83945179140231608634...  log 3  2 2 arctan
  log 1  2  dx

x 
2
0
.84006187044843982621... 
1 .8403023690212202299...

1 2

log
2
2


1
arcsin x
2 dx 
)
 x (1  x
0
2
sin 2 x
  2  2  
dx 
2
0 1  sin x
7 e  45

8
2
.84042408656431894883... 
.84062506302375727160... 

2
GR 4.521.6


 log1  x
0
2
2 
 dx
 2
k
 k !(k  7) 
k 0
 2 4 2  1 


 log1  x
0
sin  2
sinh  21/ 4
.840695833076274062...
 

2 2
 ( 3)
.84075154011728336936...  2 
 (5)
1/ 4

4
1 
 dx
 1

2
4

 1  k
2
( 1) k 1
.84084882503400147649...   k
k
k 1 3  2
2 
.84100146843744567255... 
8

1

1 .84107706809181431409...    1  
k !
k 2 

 /2
6 .84108846385711654485... 
 log 2 
2

0
4 x 2 cos x  (  x ) x
dx 
sin x
GR 3.789
2 .84109848849173775273... 

7 4

240
x4
0 cosh 2 x dx 

1 1 1

 

log xyz
dx dy dz
1  xyz
0 0 0


3 .84112255850390000063... 
111 2 341


36
216
2 (3) 

H k 3
2
k 1 k



  x cos x
2

dx 
5 .84144846701247819072...   log 2  4 G  
1  sin x
0

GR 3.791.4

=
  log1  sin x dx
GR 4.4334.10
0
.8414709848078965067...

sin 1  Imei   Imi 2 / 3  
=
 0 


1 /  
1 
J1  ,1 
2 2 

 
=

 1  
k 1


 
=
1
 cos 2
k 1


 
=

k
1 

k2 
2

4
 
=
GR 1.439.1
 1  3 sin
k 1
1

 cos x dx
1

3k 
2
GR 1.439.2
e

0
1
.84168261071525616017...   ( 7 ) 

6
coslog x
dx
x
1


1

7 
k 1 k
.841722868303139881174... 
1
1  2  log 2  
2
.84178721447693292514... 
 2  3  


dx
x

7
1

2k
 (k )  1 

k  2 k ( k  1)

 log1  x
0
1 .84193575527020599668... 
AS 4.3.65

=

(1) k


k  0 ( 2 k  1)!

1
 Ei (2)  log 2    
2
2
1 
 dx
 3

2k

2
k  0 k !( k  1)

.842048876481416666671... 
cos x

2 K 0 (1) 
1  x2
0
.842105263157894736
119 .842226666212576253...
=

dx
16
19
1
1  ( 5)  1 
log 5 x
( 5)  3  





 
 

0 ( x 2  1) dx
 4
 4 
4096 
1 
(1) k
 2 



3 J1 
;
2
;

F
 0 1
 
3  k  0 k!(k  1)!3k

 3

1
.84240657224478804816...   k
k 1 2 Fk

Bk
1
.8425754910523763415... 


3
3 e  3 k 1 3k k
.84233963038644474542... 
.84270079294971486934... 
=
(1) k


 k  0 k !( 2 k  1)

2
erf (1) 
AS 7.1.5
1
2
e

x2
dx
0
6 .84311396670851712778... 
10 .84311984239192954237... 

 
=
.8432441918208866651...

.8435118416850346340...

( 1) k

1
5
k  0 ( k  6 )( k  6 )
3
3 log 3  4 log 2  log 2 12  log 2 4 
2
1
5

 

l  l   
 6  6

k2

k
k 1 4  1
1
arctan 2 x dx
 2  log 2
G

 

2
16
4
x
0
3 3 log
 /4

 
=

0
.8437932862030881776...

16 .84398368125898806741... 
.8440563052346255265...

.84416242155207814332... 
3 1

3 1

x 2 dx

sin 2 x
GR 3.837.2
1 1 
(1  312 k )
  ( 1) k

3 4 k 0
( 2 k )!

 2k  1
2
e I0 (2)    
k 0 k  k !
cos3
2 sin
2
1

2
Berndt 8.17.8
( 1) k 1 2 k


(2 k )!
k 1

( 1) k 2 k

k  0 ( 2 k  1)!( k  1)
483 log 2 261 log 2 2 917



2
2
4

GR 1.412.4

k 4 Hk

 
=  k
k  1 2 ( k  1)( k  2 )( k  3)
1185 .84441907811295583887...  120 ( 6)  360 (5)  39 ( 4 )  180 ( 3)  31 ( 2 )  1 


 
=
 k  (k )  1
5
k 2
.84442580886220444850... 
9666 .84456304416576336457... 
 3
Li3  
 4
cosh 
2
 4k

 (2 k )!

k0
.844657620095592823802... 

 


 
=
2 (5  5 )
 


 
4 2 2

(151  75 5 ) arc csc 2 5  1
11  11 55

53 2 5

3/ 2



arc csc h 2 1  5 
Fk
  2k 
k 1
 
k

5481948
k 4 Fk 2
350 .84467200000000000000 =
  k
15625
k 1 4
1 1

5 1 1 
  , 0, 1      ,1 
.84483859475710240076... 
4 4

 4 4  4 

1
(1) k
.84502223280969297766...  cos
 
k 

k  0 ( 2 k )!

5
( 1) k  2 k 
.8451542547285165775... 
 

k 
7
k
k  0 10
(1) k

k  0 k!( 4k  1)

AS 4.3.66, LY 6.110

k3
.84515451462286429392...  9  3e  
k  1 ( k  2 )!


 (2k )
1
1
.84528192903489711771...  
sinh


2 k 1
2k
k  1 ( 2 k  1)! 2
k 1 k
2

1

(1) k ( k !) 2 4 k
2
.84529085018832183660... 
 log 1  2  
2 
8
2
k  0 ( 2 k )!( 2 k  1)

 /2
=

0
x sin x
dx
2  sin 2 x

1 .84556867019693427879... 
.846153846153846153
=
3  2 
11
13

x 2 log x
0 ex dx
Berndt Ch. 9, 32
.84633519370869490299... 
1 .84633831078181334102... 
 ( 6)

 ( 3)
7
6

 (k )
k 1
k3

   k  71


(1) k  ( k )
.84642193400371817729...  

k  2 ( k  2 )!
.84662165026149600620... 
.84699097000782072187... 
k
k 0

.84657359027997265471... 
2k
21  3 

k
k 1
1
e
2 1/ k

log 2 1


2
2
2(  e)
log x
2 x 2 

dx
 cosh(log(1  x)) x
3
1

  ( k )   ( k  2 )  
 (2)   (3)  2 
k 2
2k  1
 2k 4  k 3
k 1
2 .84699097000782072187...   ( 2 )   ( 3)


 
=
k
5
(1) k
  5
4
3
2
k 0 k  k  k  k  k  1

.8470090659508953726...
.8472130847939790866...
.847406572244788...
.8474800638725324646...


 
 3

   
  4

1
  k
k 1 2 F k
1


sin x cos x dx
0
 (1  e)  Ei (1)  eEi ( 1) 

=
 /2
2

   (1  k )

( k  1)!
1
 
=  

k 1 m 1 m !( k  1)!

1  2k 
3 .8474804859040022123...  

3 
k  1 ( k !)  k 
log 2  2 
Hk
 ( k  1)! 
k 1
k 1
 k
1 .84757851036301103063... 


( 1)k 1 e



k 1 m 1 m !( k  1)!

2
  k
k
 log
k 1
k  2

k 
2
=
 H ( x) dx
0
.84760282551739384823... 
2 .84778400685138213164... 
 4 3 1

  
2
 9

 (k  4)
k 0
k!


( k  12 )!( k  12 )!


(2 k )!
k 1

e1/ k
  4
k 1 k
AMM 101,7 p.682
Prud. 5.5.1.15

k
7 3
1
.84782787797694820792... 
  3 sin
256
4
k 1 k
1 i
1/ 4
2 Li3  e i / 4   Li3  ei / 4 
=    ( 1)
 4 4


3
k
.84830241699385145616...   3  ( 3k )  1   3
k 1
k 2 k  3

k
1
i
3 .848311877703679270993...  
Li2 (ei / 2 )  Li2 (e  i / 2 )   2 sin
2
2
k 1 k



1 .84839248149318749178... 
 
=
5 .84865445990480510746... 



.8488263631567751241...
9 .8488578017961047217...

.84919444474737692368... 
log( x  3)
dx
x2
1

  (k )


 1  ?? 

 (k )

 k (k  1) 
k 2
sum of inverse non-trivial powers of products of distinct primes

16 2

27
8

3
log x dx

x 3/ 2  1
0



1 
2 

 1  4k
k 2
97

.84887276700404459187... 

   (2k )
k 2


8 log 2

3

.84863386796488363268... 
GR 1.443.5

e1/ 4

erfi
1

2
1
54 (3)
( k )
.849320468840464412633...  
k
k 1 3

(1) k k !

(1) k
 (2k  1)!   (2k  1)!!2
k 0
1
 3
 ( 2)   
k 0
k
1 
1

 (3) k 1 (3k  2)3

6 

(1) k 1
.849581509850335443955...  
2
k  1 ( k !!)

 (k )  (k )
.849732991384718766...
  0 4
k
k 1
18 .84955592153875943078... 

.84985193128795296818... 
 H  (2k )  1 
k 1
1
1
1 




 k log 1    k log 1    log 1  2   



 1) 
k
k
k 
k 2


log 2
 (2k )  1 log 2
1
1
 

   arctan h 
4
2k  1
4
k
k 1
k 2  k


 
=

 

2 k 1
 2( k
1
2
.8504060682786322487...

.85068187878733366234... 
3 3
3

e

( 1)k

k
k  0 ( k  1)!3

1
log x
G
dx
  2
8 2 0 ( x  1) 2
1 i
1 i
2  4 log 2          
 2 4
 2 4
k E2 k
1 .85081571768092561791...  sec1   ( 1)
( 2 k )!
i
2
.8509181282393215451...  csch1 


sin i
e  e 1

1
 
= 2  2 k 1 
k 0 e

2 ( 2 2 k 1 ) B2 k
= 1 
( 2 k )!
k 1
.85069754062437546892... 

.851441200981879906279... 
 2  (2)
1 

(
1

3
)
i

4
3  2 2 / 3 

 
=
AS 4.5.65

1 1 1

  (1  i 3 )     
 2 2 2




GR 1.232.3

1/ 3

J132
1
 (2k  1) (2k  1)
k 0
2
 1/ 4
2/3

  2  2  2
 4



1/ 3
 
 
 


1
1

 H  1/ e       1    


e
k  1 k ( ek  1)

 (k )
 
=  k  1
k 2 e

( 1) k


.85153355273320083373...  e log 1  e  e   k
k  0 e ( k  1)
i  1 i
1 i
 3  i
 3  i 
.85168225614364627497... 
  
  
 
 
 
 4 
 4 
 4 
4  4 
.85149720152646256755... 



 
=
sin x
 cosh x dx 
0
1 .851937051982466170361... 
si ( ) 
4 .85203026391961716592... 
7 log 2 
.85212343939396411969... 
.85225550915074454010... 

sin x
dx , Wilbraham-Gibbs constant
x
0

 i 
 i 
2   
  
 
 2

2
2
1
log 2 
2

 k (k
k 1
2
1
 1 / 2)
1 /( 2 k 1)
(2k  1)
(2k )1 / 2 k
k 1



Berndt 8.17.8
1 .85236428911566370893... 
.85247896683354111790... 

log 2 x
1 x 2  3 dx
1 
1
   3, 3,  
4 
2
3 ( 2 )   ( 4 )  3 

 (1) k  (k )   (k  3) 
k
k 2
2k  k  2k  1
k 4 ( k  1) 2
k 2

4
1 


1  3 

1/ 3
2/3
 (1)    (1)  k 1  8k 


 
2
2

 



 
=
.853060062307436082511... 
.85358153703118403189... 
4

3
44

  log 2 

3 3

1
 k (k  3 / 2) 
k 1


 
=
 xe
x
1  e x dx 
GR 3.451.1
0
.85369546139223027354... 
2 log 2 
2
3 
k ( ( k )  1)

2 k 1
k 2


4
4k  1

2
k  2 ( 2 k  1) k

1

k
k 1 2  k
5
2
1
 1
1 1 5 
 2    2 2 F1  , , ,1 
4 2 4 
4   4


 
=
1 .85386291731316255056... 
7 .85398163397448309616... 
1 .85407467730137191843... 

 
=
.85410196624968454461... 
4 .854318108069644090155... 


0
dx
x4  1

 1
K 
 2
3 55

2
(1) k  2 k 
 

k
k  0 5 ( k  1)  k 

4 2 log 2  1 


0

( k )
16 
dx
2 

 log x log1  x

GR 4.222.3
1
k
k 2
k 1

  log 2
log x cos x
.85459370928947691247... 
 
 
dx
2 8
4
ex
0
1 .85450481290441694628... 
 (2 k  3)!
6 .85479719023474902805... 
ee  e1/ e
 ecosh 1 sinh(sinh 1) 
2

k
 3/ 2
sinh

sinh k

k!
k 1

GR 1.471.1

.85562439189214880317... 
(1) k

k
k  0 k ! 2 ( 2 k  1)

1

1

2
erf
2
bp( k )
2 .85564270285481672333...  

k
k 0 2

1 2
k 0
2k
47
6 .85565460040104412494... 
(1) k k 3
 (k )  1 
k!
k 2

k 2 e1/ k  k 2  3k  1
= 
k 3e1/ k
k 2

1 .85622511836152235066... 

 
8 .856277053111707602292... 

2 e  2

5

e
0
.85646931665632420671... 
 1  e  1 /   
.85659704311393584941... 
( 2) 
.857142857142857142
6

7
=
.85745378253311466066... 
.85755321584639341574... 
.85788966542159767886... 
coth 
(1) k

k
k 0 6
4G
coscos1
 (k )
2
k 2
.857842886877026832381... 
4
( 1) k

k
k  0 k ! ( k  )1)




.85758012275100904702... 

sin 2 x dx
x
k
1
4 (5) 

2
3

   x 1 Li2 (  x ) 2 dx
0
3
 1
 (3)  2 Li3    
 2
2

.858007616357088565688... 
S1(k ,3)
2k
k 3


cos2 x
.85840734641020676154...  4    

(1  sin x ) 2
0
1

 
=
arccos x
dx 
1 x
0

1
log 2 x
0 ( x  1)( x  2) dx 


 
=
 log(1  x
2
)
0
.85902924121595908864... 
.85930736722073099427... 
35

128
1

0
cosh x  x sinh x dx

x2
cosh 2 x
x 7 / 2 dx
1 x

24  e
2 k  ( k  3)


k!
k 0

7 .85959984818146412704... 

.859672000699493639687... 

k 1
5 .85987448204883847382... 
.86009548753481444032... 
GR 4.376.11
0

e2 k

3 
k 1 k
(k ) (2k )  1
  e
Not known to be transcendental

 2 (2k )  1
k 1
( 2 k )!


.86013600393341676239... 
k2
96  35e  
k  1 k !( k  4 )
.860525175709529372495... 

  1 1 / 3 
 2  (2) 2 / 3 
1 
1       2 1  1   



1
3
1
3
i

i




1/ 3


  2 
3  21 / 3 
2
 2  









1
(1) k 1 (3k )
 
=  3
2 k 1
k 1 k  1 / 2
k 1

4 1/ 3
k 1

e  
k 
3
k  0 k !3


 
1 .8608165667814527048...
.86081788192800807778... 
1 5
4

log
5  2 
4
1
arcsinh 
2
5
.8610281005737279228...
.861183557332564183...
.8612202133408014822...
.8612854633416616715...

k 1


1
coth  2  
4
2 2

1
1
 (2, 2, ) 
2
4

1
 (8)  
7

Fk
k
k
2
(1) k
dx
 2

x  x 1
 2k 
k 0
(2k  1)  2
k 

=

3
36


k
k 1

log x dx
x2  2
1

1

8 
k 1 k

dx
x
1
8
2
1
2
J124
1 .86148002026189245859... 
8 log 2  3 (3) 
 erf 1  1 
.86152770679629637239... 
.86213147853485134492... 
2
8


.86236065559322214...

4
 /2
4
3 log 
3


 

k cosh k
k!
k 0

( 1) k


k
k  0 3 ( k  1)


e
0



2 1  3  csc
log
k 1
k

AS 4.3.46
12

27
 (3k )
  3
k
26 (3)
k 1
11

40

sin 6 x
0 x 6 dx
GR 3.827.15

1
sin 4 k
cos 2
cos 4
3e  4e cossin 2  e cossin 4   k !
8
k 1
1/ 3



2
4
6 3 9 1
 sec1/ 3
 sec1/ 3
 sec1/ 3 
9
9
9


.8645026534612020404...

1  3 1
 3 3 
   ,     ,   
8  2 4
 2 4 
.8646647167633873081...

1  e 2 
[Ramanujan] Berndt Ch. 22
(1) k 1 2 k

k!
k 1

9



k 1
2
8
 (2)

.864988826349078353...
1
k
k 1
2
1
1/ 3

  kH k  (k  1)  1 

2 .86478897565411604384... 
x

48
.86390380998765775594... 
1 .864387735234080589739... 
2
6
3 .86370330515627314699... 
.86403519948264378410... 
x 2 dx
(sin x  cos x) 2
1

.86393797973719314058... 
(1) k 1

k 1 k!( 2 k  1)

k
.8630462173553427823...


0
1 1/ e
e  e2  e  

2e
4 .8634168148322130293...
0 0 0 0
G 


w x y z
dw dx dy dz
1  wxyz

1
1  1

  1   , 0,1 
e
2  2

20 .8625096400513696180...
.8632068016894392378...
3
1 1 1 1
4 
 2  (k )
k 1

 log 2
2
1
 3 ( (k )  1)
k 2
k
(1) k

3/2
k  0 ( 2k  1)

.86525597943226508721... 
1 .8653930155985462163...

e

7 (2)  12 (4)  18 (3)  1

=
 ( 1) k  ( k )   ( k  1)
k
3
k 2
8k 4  k 3  k 2  3k  1

k 2 ( k  1) 4
k 2


 (8k  6)


 1

k 6  cosh 4   1


k 


( 2 k )!
k 1
k 1

=
.86558911417184854991... 
.86562170856366484625... 
.86576948323965862429... 
=

1  2
2 arctan e 

 gd 1
2
arctan (sinh 1) 

 (1)
k  

 

=
k 1

1
arctan
 (k  1 / 2)


 

[Ramanujan] Berndt Ch. 2, Eq. 11.4
arcsin (tanh 1)
3


 sin  cos
2
3
6
k

( 1)  2 k 
= 

k 
k
k  0 12
.86602540378443864676... =

 

(1) k
1  5


csch
  2
2
10
5
k  0 5k  1

( 1) k
1
  2 log 1  2  

4 2
k 0 4k  1

1 
 1





8k  3 
k  1 8k  7
.86683109649187783728... 

.86697298733991103757... 

 
=

 
= 

 
= 

 
=
K135
J82, K ex. 113
1 
2 2

 2 arctan 2  2  2 arctan 2  2  log
16 4 2 
2  2




 



7
3 
3 

 cot   4 2  logsin  log sin  
 4 2  cot

16 
8
8
8
8


1
4 .86698382463297672998... 


dx

2
x  x 2
1
dx
 1 x
4
0
2
(  6 log 2 ) 
3

1
 (k  1)(k  1 / 4)
k 0

1
.86718905113631807520...  2arcsinh

2
5 .8673346208932640333...

( 1) k 1 2 k

 2k 
k 1
k2 
k


11 .8673346208932640333...

1 .86739232503269523903... 
5 .86768692676152924896... 

1  ( 3)  1 
( 3)  3  
         6 
 4
 4 
128 

0
x 3 tanh x
dx
ex
1  ( 3)  1 
x dx
( 3)  3  
         
 4
 4   0 cosh x
128 

1
 
3

sin   1  i sin   1  i csc h  2 


 (k )
k 1
Fk

k 4  2k 2  2

k 4  2k 2
k 1

(1) k

k 1 S 2 ( 2k , k )

.86768885868496331882... 
1 .86800216804463044688... 
.86815220783748476168... 

2
3/ 4

x

4
0
dx
1/ 2
13 (3)
13 (3)
1  ( 2)  1 
 5 
     ( 2)   
 

18
108
432 
6
 6 
 (1) k
(1) k 
3   

3 
( 3k  2 ) 3 
k  0  ( 3k  1)
1
 ( 3) 
3

=
.86872356982626095207... 
 (3) 
6
 MHS (4,2)
2268


(1) k
1

.86887665265855499815...   k
  log 1  k 

2 
k  1 ( 2  1) k
k 1



 ( 2k )
1 
   cosh  1
.86900199196290899881...  
k 
k 1 ( 2k )!
k 1 
1 .86883374092715470879... 
114 .86936465607933273489... 
2
1 2e
e  e 2 / e  2 

2

2 k cos k


k!
k 1
1/ k
 k  1
3 .8695192413979994957...   

k 2  k  1 



.869602181868503186151... 

 
2
2
2  1 
 1 
 1 
 1 
 
 
   (1) 
   (1)  
  
 2 


16   2 
2
2


4

2 2

cot
16
2
J312

=
Hk
2
2
 4k
k 1
1 .86960440108935861883... 
2 8 

 k
k 1

9 .86960440108935861883... 
 
2
2
 1/ 4

2
 1  sin x  dx

x
2k  (k )


k!
k 1


2
 log  1  sin x 
0
7 .8697017727613086918...
1

 e
2/k 3
k 1
GR 4.324.1

1 

2
log 2 

 
    e  x log x dx
4
2 
0
.87005772672831550673... 
3 .87022215697339633082... 
.870282055991212145966... 

 
=
.87041975136710319747... 

 
=
.87060229149253986726... 
.87163473191790146006... 
1 .871660178197549229156... 
.87235802495485994177... 
I 0 ( 2)  I1 ( 2) 

k3

2 
k  1 ( k !)
GR 4.333
k 3  2k 
 

k  0 ( 2 k )! k 

   1  2i 3 



 cos    1  2i 3  cos  5 
cos




2
2
2
3







1
1  3


3
k (k  1) 
k 1 
1
1
2 arcsin
 2 arctan

2
3


( 1) k
(2 k )!!
 

k
k
k  0 2 ( 2 k  1)
k  1 ( 2 k  1)!! 3 k

4
H ( k  1)
2
 4 log 2  2 ( 3)   3k
72
k  1 k ( 2 k  1)

1
 1/ 4
3e  2
2
(1) k
(1) k

 1 
2 
( 4 k  1) 2
8 2
k  1 ( 4 k  1)

=
1   9
 5
 3 
 7
               
 8
 8
 8 
64   8 
=
(1) ( k 1) / 2 

2
k  0 ( 2k  1)
J327


=
log x dx
x4  1
0

Prud. 5.1.4.4
.87245553646666678115... 
.87247296676432182087... 
.87278433509846713939... 
1 .8729310037130083205...


5e
1
k3

 
16 16e
k  1 ( 2 k )!

 (k )  1

k  2  ( k  1)
29
    (7)  1
20

x 2 dx
1 e x  e  x 
(1) k  2k 
 
15  3   k
k  0 6 ( k  1)  k 

.87298334620741688518... 
15
3 .87298334620741688518... 

.87300668467724777193... 
 ( 1)  (3k  1)   (3k )   (3k  1)  3 
k 1
k 1

 
=


1
2
 3
2 k 1 k  1

k3
2 .87312731383618094144...  4e  8  
k  1 k !( k  2)

k4
9 .87312731383618094144...  4e  1  
k  1 ( k  1)!
10 .87312731383618094144... 

4e 

 ( k )  1
1 
 

 (k ) 
k 2 
1
log 

1 .8732028500772299332...

.87356852683023186835... 
2 ( k )  1
 (k )
k 2


J153





1 

1 .87391454266861876559...    1
 2k  
k 1
  
  k 
  
2 .87400584363103583797... 
7 .8740078740118110197...

.87401918476403993682... 

 3   log 2 2

8 2
62
log 2 x
0 2 x 2  1 dx 
1
 1
2  
6 2  4 
 /2
 sin
0

log 2 x
0 x 2  2 dx 
 /2
3/ 2
x dx 
 cos
3/ 2
0
xdx 
GR 3.621.2
 (k )

.87408196435930900580... 
 (2 k  2)!

k 2
.87427001649629550601... 
.87435832830683304713... 
1

  k cosh
k 1

1 i 
1 i 

  2  i   
  2  i 
2  4 
 4 

1
H
3
3  3
   2k k 
 3 log  3 log
4
2
3  3
k 1 3
 (k )
1
 
k
k 2 n2 n
ω a nontrivial 

.87446436840494486669... 

.87500000000000000000

 
1 .87500000000000000000
3 .87500000000000000000
=
=
.87578458503747752193... 
1
3G

7

8

15

8
 1  k (k  5 / 2) 
31

8
3 .87578458503747752193... 
=


k 1



8
3
 (2k  1) (2k  1)  1 
k 1


 e
k 1

1
9

k

3
2 
8
3

0
k

 (1) k

log 2 x
0 x 2  1 dx 

x2
0 cosh x dx 
 2 i Li3 (i )  Li3 ( i ) 
4
Prud. 5.3.1.1
x 2 tanh x
dx
ex
1
 Re(1)1 / 6 
2

(1) k
 
= 

k
k  0 ( 2k )!4
2 .87759078160816115178...   G


( 1) k 1
.87764914623495130981... 
 log 2   2

2
k 1 2 k  k
.87758256189037271612... 

integer power
(1) k


k
k 0 7

Hk
= 
k 1 ( k  1)( k  3)
=

   (k ) (k )  1   
k 2
.87468271209245634042... 
1 1
 
k k
GR 3.523.5

x2
 e x  e x

cos
AS 4.3.66, LY 6.110


 
=
  Lik (i )  Lik (i )  
k 2
1
 
=
dx
 log1  x  x 
2
0
GR 4.295.2

=

1
 log1  ( x  1)
2
0

 dx 

 1  x2 
=  log
 dx 
 1  x2 
0
1
 
 /2

 
=

x dx
 1  cos x 
Prud. 2.5.16.18
0
1 1


=
0 0
1 .87784260817365902037... 
x y
dx dy
2 2
y
 1 x
 coth 

1 .8780463498072068965...

109 e1/ 3

81
k4

k
k 1 k ! 3
5 .8782379806992663077...

3 
1
  2 , 4 ,  
8 
2

log 3 x
1 x 2  2 dx

1
( 1) k
1
 4 cos
4  

.878429128033657480583...  2 2 sin
k
2
2
k  0 ( 2 k )! 2 ( k  1)


1 
1
k  1  ( 2 k  1)
.879103174146665828812...   ( 1)
  log 1  2 

k
k 
k 1
k 1 k

log k
.87914690810034325118...   2
k 1 k  1

1
.87917903628698039171...  

Related to Gram’s series
k 1 k!k  ( k  1)

.87919980032218190636... 
.87985386217448944091... 
.8799519802963893219...

2k

k
k
k  0 2  ( 1)

k!

k
k 1 k

 
 0 (k )
1




k
j k
k 1 3  1
k 1 j 1  3   1

=


1
 3
i 1 j 1 k 1
ijk

( 1) k
2 J1 1  
k
k  0 k !( k  1)! 4

.88010117148986703192... 

1 .880770870194...

 0 (k )
 k (k  1)
k 1


 
=


j 1 k 1


1
 k  
k 1
1
jk ( jk  1)
1

  1    

k 
4 .880792585865024085611... 
I 0 ( 3)

1  tanh 1
e
( 1) k
.88079707797788244406... 

  2k 
2
e2  1
k 0 e

1 1
(k!) 2

5 .88097711846511480802...  2 F1  2, 2, ,   
2 4  k 1 (2k  2)!

.88107945069109218989... 
1 .88108248662437028022... 
J994

1
(1) k 2


;
4
,

6


0 F1 
k
2

k  0 k!( k  3)!2

1
 (k  1)(k  5 / 6) 
9 log 3  12 log 2  3 3 
k 0


 
=
 (k )
6
k 2
1 .88109784554181572978... 
 
cosh2
2


1

2
 k 1/ 2
k 0
 1 i 
1  i 
= i  
  
 
 2 
  2 
2 .88131903995502918532... 

k 2
 tanh

k
2
( 1) k  2 k 
 

k
k  0 4 ( 2 k  1)  k 

.881373587019543025232... 
=


log(1 + 2) = arcsinh1 
log tan

8
 log sin
3

 log sin 
8
8
1
  i arcsin i 
2

  k  12 
.88195553680044057157...   
2 
k  1 2 ( 2 k  1)

1
1 .88203974588235075456...   2
k  1 k log( k  1)
 
.88208139076242168...
.8823529411764705
=

arctanh

2
15
=
17
erf 2 

.88261288770494821175... 
e
1/ k !
k 2
(1) k 2 2 k 1

k  0 k !( 2 k  1)

 1
1  3
.88261910877658153974...   
csc 3 
3
6
10 .88279618540530710356... 
2 3 


 log1  x
0
(1) k


2
k 2 k  3

63 
 dx
 1
2
J85

.883093003564743525207... 
3 .88322207745093315469... 
2k


k
k 1 4  1
  5  1 


 log1  x
2
0
4 
 dx 
 1

.88350690717842253363... 
 (2k  1) (2 k )  1 
k 1
3 .8836640437859815948...


e  1  Ei ( 1) 
Hk
 ( k  1)!
k 1
(1) k 1
HypPFQ1,1, 2, 2, 2,  1  
2
k 1 ( k!) k

.88383880441620186129... 
4 3
.88402381175007985674... 
 g3 
81 3
64 5
1 .88410387938990024135... 
volume of the unit sphere in R5
10395
1 .88416938536372010990... 
1 .8841765026477007091...

e log 2 
11 3


4
2

x
7
2
e  x log x dx
0

.88418748809595071501... 
=
Hk k ! k !
k  1 ( 2 k )!

2  (1)  1 
 2
     (1)     3 (2  log 3) 
27 
 3
 3


 1 i 3 
 i 3  1
2  2  log 3
  i Li2 


 iLi2  

 2 
3
2
3 3  3





16
16
1
k !k !

arcsinh   ( 1) k
.88468759253939275201... 

17 17 17
4
(2 k )!4 k
k 0
=
2G   
1 .88478816701605060798... 
2
1  cosh 3 
3
1 .88491174043658839554... 

 log 2

2
3 .88473079679243020332... 
1 .8849555921538759431...

.88496703342411321824... 


Hk 2 k ( k !)2

(2 k )!
k 1

3k
3 sinh 3   k

k  0 2 ( k  1)
1
log 2 x dx
2
 
G
2 2
32
0 (1  x )
3
5

1
2
k2

  2
2 
12 16
k  2 ( k  1)

J310


 
=
 k  (2k )  1 
k 1
1 .885160573807327148806... 
 2 log 2 2


6
2

log 2 x dx
0 ( x  2)2

25 .88527799493115613083... 
2 k kHk
e 1    2 Ei (2 )  2 log 2  1  

k!
k 1
2
3
1 .88538890738563...


2
( x) dx
2
2 .885397258254306968762... 
.88575432737726430215... 
2
6
 4 log 2  log 2 2 
12
2 (2)   (3) 


 
=
2 Hk
 k (k  1)
k 1


 
=

1
 
=
0
x
x log 2 x
0 ( x  1)2 dx 
 1
2
x 2 dx
  x x
x
 2
0 e e  e
 8
1 .88580651860617413628... 
5 15 log 2 5



4
4
8

3
x dx
0
1
8 .88576587631673249403... 
.8858936194371377193...
2

x 2 dx
 e
 arccos x log

log 2 x
dx 
x( x  1) 2


2
1
5k  (k )  1

4k
k 2


35
16
5 .8860710587430771455... 
 ( 4 , 1)
e
2 .88607832450766430303...  5
.88620734825952123389... 
.88622692545275801365... 

 

(1) k 1 32 k 1
erf 3  

2
k  0 k !(2 k  1)

 

 

2
 3
=    
 2
( 1) k 1 ( k  12 )! 3k
= 

k!
k 1

k  1/ 2 

=  1 

k  3/ 2 
k 0 



AS 6.4.2
AS 6.1.9
sin 2  x 2 
0 x 2 dx 

=

=
e
 x2
dx 
LY 6.29
0

=
e
 tan 2 x
0
sin x
dx
x cos 2 x
GR 3.963.1

x4
dx
.88626612344087823195...  24 (5)  1   x x
0 e  e  1
1
24 .88626612344087823195... 
.88629436111989061883... 
1 .886379184598759834894... 
.88642971056031277996... 
2 .88675134594812882255... 
.88679069171991648737... 
.8868188839700739087...

.887146038222546250877... 
24 (5)   
2 log 2 
(4)
log 4 x
1  
dx
1 x
0
1
2


5  2
2

cot  2   2
  2 k  (2 k )  1
2
2
k 2 k  2
k 1
k

(1)

k 1
k 1 k
5
3

1
2

( 1) k

csch
  2
2
3
3
k  0 3k  1

E2 k
1
2
sech  1/ 2

AS 4.5.66

 1/ 2
k 
2
e e
k  0 ( 2 k )! 4
1 i 3
1 i 3

  log  
 log  
 2 
 2 

1

log 1  3  


k 
k 1

log 4 k
24 .8872241141319937809...  
k  2 k ( k  1)
 
=
9 .88751059801298722256... 
9 log 3
(1) k  1

k
k 1 3  2

Fk 2
2 .88762978583276585795...  

k 1 k !

.88760180330217531549... 

 ( 1)
k 1
k 1
 (3k )
k
.8876841582354967259...
1  
( 1) k

 2   log 2  Ei (  )  

2  k  0 ( k  1)!2 k ( k  1)

38 .88769219039634478432... 
 k  (k )  1
5
2
k 2
3 3

6
.88807383397711526216... 
CFG D14

8 .8881944173155888501...
79



 
.88888888888888888888
=
1 .88893178779049819496... 
8

9
(1) k

k
k 0 8


3 2

5
25   2 

7 .8892749112949219724...

 ( 1) k 1
  ( k  1 / 5)
k 1
2

( 1) k 1 

( k  1 / 5) 2 
1
 k ! (k  1)  1
k 1

.8894086363241...



1
 1  k ( k  2)( k  4) 
k 1

1 .89039137863558749847...   4 Li3  

.89045439704411550353... 
5 .89048622548086232212... 
.890729412672261240643... 
1 .89082115433686314276... 
.890898718140339304740... 
1

2
 2  3 3 
81

x 2 dx
0 ex  1 / 2

log x
dx
( x 3  1) 2
0
 
15
8
  2 log 2 
3
 5
log 3     
 6
2
1 
1
   2, 3,  
4 
2
2
 1/ 6

log 2 x
1 x 2  2 dx 

(1) k 
  1 

6k  1
k 1 

1
.8912127981113023761...

 445  199 5 
1 5 
2 
 4 2 
  log

 32 (85  38 5 )   5 log 56  24 5 
80 
5 


k

1   3
(1)
 1 
         
=
10   5 
 10   k  0 5k  1
.88831357265178863804... 

1 log 2 x dx

HypPFQ [{1,1,1},{2, 2, 2},1]  
2 0 ex
(1) k 1

2
k 1 k!k
(1) k

= 
3
k  0 k!( k  1)


 
.89164659564565004368... 
.89169024629435739173... 


3   5
  
8  2
x 4 dx
e
x2
0
1
 /2

1  e 2    cos2 (tan) dx 

4
GR 3.716.10
0

=
cos 2 x
0 1  x 2 dx
 (k )

.891809...

 k (k  1)
k 1

.89183040111256296686... 
x
1
2
dx
log(1  x )
5
.89257420525683902307...  4 log

4
.89265685271018665906... 
1 .8927892607143723113...


(1) k
1


4


k
k
k  0 4 ( k  1)
k 1 5 k


e
0
x
dx
 1/ 4
 1 / 3


 1 / 2
3 log 3 2

3 .89284757490956280439... 
.892894571451266090457... 
ek
k
0 k!2

1


p 
k  2 p prime k
ee / 2  
  ( p)  1 
p prime
 
1 1
 log 1     in Mertens’ Theorem HW 22.7

p  p 
p prime 


log  (k )
=     (k )
k
k 2
2i
2 i
.89296626185090504491...   Li4 e  Li4 e
.892934116955422937458... 
B1   

 
 
1  1
 4
.89297951156924921122...         
 3
3  3
.89324374097502616834... 

 
=

e
 x3
dx
0

 3 3 1 
HypPFQ1 ,  , ,  
2 2 4


2
 /2
H0 (1) 
 sin(cos x) dx
0
(1) k

2 
k  0 (( 2 k  1)!!)


2 .89359525517877983691... 

0
1 .89406565899449183515... 
x 2 log(1  x )
dx
ex  1
7 4  H ( 2 ) k


360 k 1 k 2
1

 
=
 log(1  x)
0
log 2 x
dx
x
i

 i
2 cos  e  / 2   i i  (i ) (  i )
2

3

1
 1
2
.894374836885259781174...  8G   
  ( 3)    14 ( 3) 
 4
2
96
1 .89431799614475676131... 
2
.8944271909999158786... 

5
17
.894736842105263157
=

19
.89493406684822643647... 
=
=
1 .89493406684822643647... 
2
1 .89511781635593675547... 
.89533362517016905663... 
.89580523867937996258... 
.89591582830105900373... 
 2k 
 
k

2
1
 
6
4
1
(1  x  x 2 ) log x
dx
0
x 1
 ( k  1)  1

e1/ k
  2

( k  1)!
k 1
k 2 k

1
Ei (1)  li ( e)    
k 1 k ! k

si ( 2 )
(1) k 2 k
 
2
k  0 ( 2 k  1)!( 2 k  1)

i
sin k
i
i
Li4 ( e )  Li4 ( e )   4

2
k 1 k
 2
csc

4
2
k 1
( 1)


2
k 1 2 k  1

=
k2

4
k 1 (k  1 / 4)


3
2k  1
 

( 1) k k ( ( k )  1)

2
6
4
k  2 k ( k  1)
k 2


1 
 1
 3
    ( k )   (2 k )
 2


k  k2 
k 2 k  k
k 2


k2  k 1
2

3
3  
2
k 1 k  2 k
k  2 ( k  1) k ( k  1)

.895105505200094223399... 
( 1) k

k
k  0 16



1
2
AS 5.1.10
.89601893592680657945... 
.89603955949396541657... 


1 

  1 

16k 2  1
2 2 k 1 

log( x 2  1)

2
log 3 
 
dx
9
x2  1
3
0
sin
1
.896488781929623341300... 
x
x2
dx
0
.89682841384729240489... 

 1
 1

 2 Li2        , 2, 1 
 2
 2

 1
log 2 2  2 log 2 log 3  log 2 3  2 Li2  
 3
 
=
 
( 1) k
=  k
2 
k  0 2 ( k  1)




1 .89693992379675385782... 

 ( k  3)
k !2k
k 0
2
.89723676373539623808... 
1 .89724269164081069031... 
.89843750000000000000
e
x
0
x
 12

e1/ 2 k
  3
k 1 k

11
7


3 1
6 2


dx
 1 x
12 / 7
0
115
1

=
   ,  3, 0  
128
5

.89887216380918402542... 
k3

k
k 1 5

1  log 2 k


2 k 1 k ( k  1)
( 1) k  2 k 
 .8989794855663561964...  2 6  4   k
 
k  0 8 ( k  1)  k 


4 .8989794855663561964...


.89917234483108405356... 
.89918040670820836708... 
.89920527550843900193... 


9 .8994949366116653416...

5 .899876869190837124551... 
24  2 6 
3
1
erf

2
3
( 1) k

k
k  0 k ! 3 ( 2 k  1)

1


2
cot
  2
2
2 2
k 1 2k  1 / 4

3

 17

tan

2
4
17
98 
3 (3) 

k
k 1
2
1
 3k  2
17 4 5 2 1



360
24
4

H k H k 2
k2
k 1

J1064
11 .89988779249402195865... 
1
1 G


.90000000000000000000
=
.9003163161571060696...

 
=
9

10
2 2

(1) k

k
k 0 9


1 

  1 

16k 2 
k 1 
 0 


 1 / 4
  

k 2 



 
=
 cos  2
k 1
2 .900562693409322466279... 
GR 1.439.2
3 ( 3)  2 

  log 2 
16
24 2
1
 arccot x log
2
x dx
0
 2 (2k )

1 .90071498596850182787... 
GR 1.431

2k
1
2 .90102238777699398997...  
k 1 k ! 1 / 2
k 1

30 .90110011304949758896... 
11e  1 

k5

k  1 ( k  1!
 1
.90128629936044729873... 
K   
  2
2
2
 2k  (1) k
 

32k
k 0  k 

.90135324420067371214... 
1 si ( 2)

   log x cos2 x dx
2
4
0
.90154267736969571405... 

(1) k 1
3 (3)

  (3)     1,3,0   
k3
4
k 1
1
1

 
=
  log(1  x)
0

=
.901644258527509671814... 
1 .902160577783278...
GR 4.315.1
x 5dx
e
0
log x
dx 
x
AS 23.2.19
x2
1
 1 4 1
 
2 F1 1, , , 
 3 3 2
(1) k

k
k  0 2 (3k  1)

 Brun’s constant, the sum of the reciprocals of the twin primes:
1

p atwin p

prime
Proven by Brun in 1919 to converge even though it is still unknown
whether the number of non-zero terms is finite.

.902378221045831516912... 
 ( 1)
k 1
k 1
k 2 ( k  1)
2k
.90268536193307106616... 
.9027452929509336113...

 cos 2
2  2
5
     
3  3
 3

2 .90282933102514164426... 
 ( 1) 
k
k 2

1 .902909894538287347105... 
3
( k )  1 
I k (1)

k 0 k!


1  2 log k
2 k 2 log2 k   x  ( x)  1dx
2
e
.9036746237763955366...   sin
  Imi e 
2

  (k )

 1
.90372628629864941069...   

k  2   ( k  3)

1 .9036493924...





463
1

  ,  5, 0 


512
5
3 2
7 ( 3)
.9043548893192741731... 
 log 2 

8
4

(1) k k 2 ( k )
 
= 
2k
k 2
6 .90429687500000000000



3
(1) k
log 2  3  
6
6
k  0 6k  1

1
.90406326728086180804...   k

k 0 3  1
2  3 4  2 2 / 3  2 
 2 
1 .904148792675624929571... 
e  I 0    I1    
1 F1  ,2,  
3  2 3 3
 3 
  3

k  2k 
 
 
= 
k 
k  0 k !3  k 
.90377177374877204684... 
=
6

k5

k
k 1 5

16k 2  6k  1

3 
k  2 k  2 k  1
( 1) k

.9045242379002720815...  
k  0 (2 k )!( 4 k  1)


 2 
1 dx
   cos 2  2 
 
=
C 

2   1
x  x


2k
9 .905063057727520713501...  8 (3) 
  3  (k  1)  1
2
k 1 k
e3  1
.90514825364486643824... 

e3  1
J974

J148
4108

1
16 .90534979423868312757... 
   ,  5, 0  
243

4

 (k )  1
.90542562608908937546...  
Fk 1
k 2
1 .90547226473017993689... 


k5

k
k 1 4

e
si ( 2)
( 1) k

1

k  0 k !( k  2 )( 2 k  1)

kHk
( 1) k 1 Hk
2
.90584134720168919417...  2 log 2  log 2   k
 1 

( k  1)( k  2)
k  1 2 ( k  1)
.90575727880447613109... 


 
=
 4k
k 1

.90587260444153744936... 
.90609394281968174512... 
12 .906323358973207156143... 
Hk
 2k
1
2
 2  (2k  1
k
k 1
6 .90589420856805831818... 




  (k  1)(1k  1 / 6)
3
 3  3 log 3  4 log 2 
5
e

3
2
8 (3) 

 2 k  (k  1)  1

3
k 0
k
2
k 1
5 1 1
.90640247705547707798...       
 4 4  4


 
=
e
x4
dx
0

.90642880001713865550... 
1
 k ! k
k 1
2
k
1 1
  (1)  2k 
  
.9064939198463331845....  2 F1  , ,2,1   k
2 2
 k  0 16 (k  1)  k 
.9066882461958017498...

.9068996821171089253...

=


2

2 3
log x sin x
dx 
x
0


3
sin

3



1
(1)
k
 (k  1)(3k  1)
k 0
1

  6k  1  6k  5 
J84, K ex. 109e
k 0
maximum packing density of disks

=

 
CFG D10
( 1) k


k
k  0 3 ( 2 k  1)


dx
dx
  4

=  2
2

3


1
x
x
x
0
0

=
 

=
x
4
1
 
=
x
0
1 .90709580309488459886... 

dx
2
1
 3x
0
2
0

J273
2
x dx

 x2  1
1

log 1  3  dx

x 
1
 3

cosh
4
2



3
 1  (2k  1)
k 1

2

 


3
k2  3/ 2
  2
2
2
k 0 k  1/ 2


(k )
b( k )
.90749909976283678172...   2   ( 2) 
log  (2 k ) 
k 1 k
k 1 k
8 .907474904294977213681... 
3 csc h
sinh 
Titchmarsh 1.6.3

.90751894316228530929... 
  (k )  1 log k
k 2
23 .90778787385011353615... 
5 5

64

x4
0 ex  e x dx 
 /2

 
=
  log tan x
4
dx 
GR 4.227.3
0
1

 
=
.907970538300591166629... 
.90812931549667023319... 

 
=
.90819273744789092436... 
log 4 x
0 1  x 2 dx 
5 2 log 2 2
1 i 
1 i 

 Li2 

  Li2 
48
4
 2 
 2 

1
4 2 
cos( k / 2)




 
90 48 96 768
k4
k 1
1
Li4  ei / 2   Li4  e i / 2 
2
 5  i 3
 5  i 3 
1
 2   
  
 
3
 2 
 2  




k 1

 (3k  1)   (3k )  2 


3
k 2 k  1
k 1

5
1
 4
 
=
6 k 1 k  k

1 (2)  4 
k
( 1)  4 
.90857672552682615638...          
3
 3 6
 3
k  1 ( k  1 / 3)


GR 4.263.2
 
=
GR 1.443.6

.90861473697907307078... 

 0 (k )
 log(1  2  k )
 

k
k 1

k
2 k
k 1
 (1) 2

.90862626717044620367... 
k
(k )
2


 2
j 1 k 1
1
jk
jk

k 2
.90909090909090909090
=

.90917842589178437832... 

k 2
.9092974268256816954...

(1) k

k
k  0 10
1
k !log k

10

11
sin 2 
(1) k 2 2 k 1


k  0 ( 2 k  1)!

1  e(cos1  sin 1)
.90933067363147861703... 

2
3 log 3  2 log 2 
AS 4.3.65
e
 sin log x dx
1
1
1 .90954250488443845535... 
2 k sin( k / 2 )


k!
k 1


2
 log1  x  dx
0
1 .90983005625052575898... 
1 .90985931710274402923... 
1 .91002687845029058832... 
.91011092585744407479... 
.91023922662683739361... 
.91040364132111511419... 
.91059849921261470706... 
.91060585540584174737... 
1 .910686134642447997691... 



5
3  5  1  cos 
2
5
6


1
k
k 0

5
 (4) 
4


 (2k )
 1 
Li2  2 



k 2
 2k 
k 1 2 k
k 1
1   ( 2)   (3) 
1

log 3


Hk 1
3
k 1 k

dx
3
J153
x
0

1 
1

 coth 2    2
8 4
k 0 k  4
sin(log  )  Im i 
 4
Li3  
 5
1

2
e
 1 
erf 
 
 2
2

k !2k k

k  1 ( 2 k )!

244 .91078944598913500831... 
8 3
 96  64 log 2    x  3/ 2 Li2 (  x ) 2 dx
3
0

.911324776360696926457... 
 2k  2 

 k 
 5 (k  1) 
  (k )  (k )  1 
k 2



k 2 n2
 (k )
nk
.91189065278103994299... 
.91194931412646529928... 
1 .911952134811735459360... 
23 .9121636761437509037...

.91232270129516167141... 
1 .9129311827723891012...

.91339126049357314062... 
.9134490707088278551...

3 .91421356237309504880... 
.9142425426232080819...

9
2
3
34

 11
 7
1
2


  1  2
sech
cosh

k  k  2
2
2
3
k 1 
65
 (5, 1)
e
1
 19  14 2 csc( 2 2 )
112
3
7

1

k  1 2 k 
k
 

 ( 3) 


2
5
 2 
2
2
1


arctan

7
7
7
CGF D4

x
0
2
dx
x2

32
1
  ( 4 ,1 / 2 )   k
35
k  0 4 (k  4)
k

3
2 .91457744017592816073...  cosh 3  
k  0 ( 2 k )!
1
1
.91524386085622595963... 
cot 
2
2
i
i  e  1
i  cos1  i sin 1  1
 
=


 
2  cos1  i sin 1  1
2  ei  1
.9142857142857142857

=


sin k
a
a0
k 1 k

(1) k
.91547952683760158139...  
k  0 7k  1
 
=
lim 

k4

k  1 k !( k  5)


1
3 .91585245904074342359...   3 tanh
  2
2 3
k 0 k  k  1 / 3
4 .91569309392969918879... 

120 ( 46e  125) 
( 1) k B2 k


(2 k )!
k0

.91596559417721901505... 
=
G   (2)  Im Li2 (i) , Catalan’s constant
 log 2
8

LY 6.75
i  1 i 
 1  i 
 Li2 
  Li2 
 
2  2 
 2 
(1) k

2
k  0 ( 2 k  1)

1
1




2 
2
(4 k  1) 
k  1  ( 4 k  3)

=
=
=
=
=
=
1 
( k !) 2 4 k

2 k  0 ( 2 k )!( 2 k  1) 2
1  (3k  1)( k  1)
 ( k  2) 

16 k 1
4k
1  k 
3

k   k  1,  
8 k 1 2 
4
3 
1

log 2  3  
8
8 k  0  2k 
  ( 2 k  1) 2
k


=
e
1

0
Adamchik (28)

log x
dx 
x2  1
1 x
1
=
Adamchik (27)
x
dx
 e x
x
0
=
Adamchik (23)

log x
dx
2
1
x
1
dx
 log 1  x  1  x
2
GR 4.297.4
0
=
 1  x  dx
  log

 x  x2  1
0
Adamchik (12)
=
 1  x 2  dx
  log

 2  x2  1
0
Adamchik (13)
=

1
1
 /4
 log tan x dx
GR 4.227.4
0
=
1
2
 /2

0
x
dx
sin x
Adamchik (2)

=
1
x sechx dx
2 0
Adamchik (4)
 /2
 arcsinh(sin x ) dx
=
Adamchik (18)
0
=
 /4
 /4
0
0
 2  log(2 sin x ) dx  2  log(2 cos x ) d
1
=
arctan x
0 x dx 
Adamchik (5), (6)

arctan x
dx
x
1

GR 4.531.1

arctan x
dx
1  x2
0
=

=
1
K ( x 2 ) dx
2 0
=
1
   E ( x 2 ) dx
2 0
1
Adamchik (16)
1
1 1
=
  (x  y )
0 0
.9159981926890739016...

5 .91607978309961604256... 
.91629073187415506518... 
.91666666666666666666
=
=
.9167658563152321288...

Adamchik (17)
1
dx dy
1 x 1 y
 (8  3  18 log 2)
12 2
Adamchik (18)

log(1  x 4 )
  4
dx 
x (1  x 4 )
0
35

5
3k
 3
 Li1     k 
2
 5
k 1 5 k
k

(1)
11

12 k  0 11k


k5  k4  k2  k 1
1




4
3
7
(k  k )(k  1)
k 2
k 2 k  k
log
cos 4 2 cosh 4 2 
(1) 2
k  0 ( 4 k )!

k


1 3  1 
HypPFQ  ,  , ,1,  
 4 4  64 


k3  k 1
.91767676628886180848...  2   (4)   5

4
k 2 k  k
.91775980474025243402... 

.91816874239976061064... 
k 2
(1) k  2 k 
 

k  0 ( 4 k )!  k 

log(1  x )
  3(log 3  1) 3   
dx

9
x (1  x )
3
3
0
 6
 
 5
k 1
3
1
k
k
.91681543444030774529... 


3


1
(1) k
 1 
 1 / 4 J1 / 2 

 2  k  0 ( 2 k  1)! 2 k
2
2

7
13 1/ 4
k !k 2
 1
.91874093588132784272... 

e
 erf    
 2
16 32
k  0 ( 2 k )!
n
n!e
.91893853320467274178...  log 2  lim n 1 / 2    (1) 
n n
.91872536986556843778... 

 
2 sin
= Stirling’s constant

,
GKP 9.99
log n
1
the constant in log n!  n log n  n 
 
 ...
2
12n
1

 
=
 log  (x ) dx 
GR 6.441.2
0
1

 
=
 1
x
x
dx
  log x  1  x  2  x log x 
GR 4.283.3
0


sinh 
1
   (4 k )  1

.9190194775937444302... 
   1  4   exp  

 k 1

4
k 
k
2 
1
 
=
2 2  i   2  i 
 7
.9190625268488832338...    
 4
.91917581667117981829... 
 (1) (i ) (1) (i )

2 .91935543953838864415... 
Hk 2


k 1 k !
1
.9193953882637205652...  2  2 cos1  4 sin

2
2
.92015118451061011495... 
 
 2
2

 
e
1

   ,  1, 0  
2
(e  1)
e


(k )
=  k
k 1 e  1
.92068856523896973321... 
2 .92072423350623909536... 
2
e1 / 8
erfi
 log 2
2

sin 2 x
 
dx 
2
0 1  sin x
.920673594207792318945... 

( 1) k

k  1 ( 2 k  1)! k

1
2 2


k 0
0
k
e

 log1  2 x
1
2

 dx 
 1

k
(1) k k!

k
k  0 ( 2k  1)!2

1
 2G   
0
log(1  x )
1  x2
dx
GR 4.292.1
1
=
arccosx
dx
1 x
0

GR 4.521.2
 /2
=
x sin x
 1  cos x dx
GR 3.791.9
0
12 .92111120529372434463... 
.92152242033286316168... 

(1) k
2  8csch   2
2 k 0 k  1 / 4
2

 sin  2  sinh  2 

2 2 cosh  2  cos 2





1
1  4
 1   ( 1) k 1  (4 k )  1
k 2 k  1
k 1

Hk
2
1 .92181205567280569867...  4 log 2  
k 1 k ( k  1 / 2 )
2

e I 0 ( 2) 1
1  2 k  1
 
7 .9219918406294940337... 


2
2 k  0 ( k  1)!  k 
 
=
.92203590345077812686... 

 
=
.92256201282558489751... 
2



1

16

cos( k / 2 )

k2
k 1

6
4
1
Li  ei / 2   Li2  e i / 2 
2 2

1
( 1)1
 erf  
k
2
k  0 k ! 4 ( 2 k  1)


1
.92274595068063060514... 
 ( x  1) dx
0
.92278433509846713939... 
3
    ( 3)
2

1 .92303552576131315974... 
 
k 1
2
( 2 k )  1

12
(1) k
 
k
13
k  0 12

(1) k 1 ( 4 k )

.92311670684442125248...  
( 2 k  1)!
k 1
.923076923076923076
=
.92370103378742659168... 

.923879532511286756128... 
.92393840292159016702... 

1
k
k 1
2
sin

 4  1
tanh

2
4  1
1
k2

k
k 0
2
1
 k 

3
2 2
 cos  sin
8
2
8

90
1
 (k )

  4
4
 (4)
k 1 k
x
.9241388730...
 point at which Dawson’s integral, e
x2
e
0
t2
dt , attains its
maximum of 0.5410442246.
AS 7.1.17

4 log 2
( 1) k
 1  2

.92419624074659374589... 
3
3
k  1 27 k  3k

Hk / 2
=  k
k 1 2

(1) k
.924299897222937...
 
k  2 log k
.924442728488380072202... 
2
25

.92465170577553802366... 
cot

5
(1)
csc


 
5
0
Berndt 2.5.4
log x
dx
x5  1
k
 8k  1 
k 0

5 5 log 3
5
5k  (k )  1


 

3
2
3k
6 3
k 2
log 2
1
.92491492293323294696... 
  log( 3  2 2 ) 
3
3 2
k

( 1)
 
= 
k  0 ( k  1)( 4 k  1)


k 2 (2k )
k 2 (4k 2  1)
3 2
.92527541260212737052... 

 

4

2
3
32
4k
k 1
k 1 ( 4 k  1)
5 .924696858532122437317... 





 
=
.925303491814363217608... 
arctan x
dx
1  x2
1

 erfi (1)  2 
( k  12 )!

1
k 1 k !( k  2 )!

sec1
1
 i
2
e  e i
1  csch1
e 2  2e  1
.925459064119660772567... 


2
2e 2  1
.92540785884046280896... 
(1) k
=  2 2
k 0 k   1


 
.9260001932455275726...

1
2
sinh 
2
 
=
.92655089611797091088... 
k
1
2
4
2
cosh 2 

1
  (k  i )
k 1
2k  2
 2k 2  1
k 1
.92614448971326014734...  3  2 (5)




2

1 

( k  i )2 
Prud. 5.1.8.9
.92684773069611513677... 
1
8 .92697404116274799947... 
.92703733865068595922... 
.927099379463425416951... 
2
8
k ( k )

2k
k 2

 log 2 

1 2
  log2 2 
12
 
2

log 3 x
0 (x  2)(x  1) dx 
 1
1 1 5 
 2    2 F1  , , ,1 
4 2 4 
8   4


GR 4.262.3
dx
x4  1
1
3 log 2 3
 1
 6 log 2 log 3  6 log 2 2  3 Li2  
 4
2
( 1) k
=  k
2 
k  0 3 ( k  1)
.92727990590304527791... 
4k  1
k 1
1



 2k (2k  1)
 1 

e
x
0

x
 13
5  3 5  5  5 5  3 5  5  5




10
10
 2 
 2 

2k  1
 
= 
  (1) k Fk 1  ( k )  1
2
k 2
k  2 k ( k  k  1)

1
4
1
.927295218001612232429...  2 arctan  arcsin 
 arctan 
7
5
4
2

i
i
(1) k


 
= i log1  i log1     k
 2
 2  k  0 4 (2k  1)
7  (2)

.92753296657588678176... 
4
2

1
cos( k / 2 )
i/2
i / 2
.927695310470230431223... 
Li3  e   Li3  e   
2
k3
k 1

4
 (8)
 (k )
.92770562889526130701... 

  4 
105  ( 4 )
k 1 k




2 .92795159370697612701... 

 
=

HW Thm. 300
e2 cos 1 sin( 2 sin 1)  sin( 2 sin 1) cosh( 2 cos1)  sinh( 2 cos1) 


i 2 ei
i
e  e2 e 
2


2 k sin k

k!
k 1

1 .92801312657238221592... 
4

2 (1  log 2 )    log 1  2  log x dx 

x 
0

3 .928082040192025996...


k 1
pd ( k )
k!
( 1) k  2 k 
 

k
k  0 12 ( k  1)  k 

.92820323027550917411... 
4 36 
 2k  k 2
  k

k 1  k  6

6 .92820323027550917411... 
48  4 3 
GR 4.222.3
.9285714285714285714
(1) k

k
k  0 13

13
=

14
1 .92875702258534176183... 
2
49


3
36
(1  x 3 ) log x
0 x  1 dx
1

1
2k
.92875889011460955439... 
I (2 2 )  
2 2
k  0 k !( k  2 )!

sin 5k
5 
i
1  e5i
.92920367320510338077... 
 log
5i   
k
2
2
1 e
k 1
4 .92926836742289789153...   e 
.92931420371895891339... 
 (2)   (3)   (4)  3 
k2  k 1

k4
k 2

k2  k 1
3 .92931420371895891339...   ( 2)   ( 3)   ( 4)  
k4
k 1


 ( 3k )
3
1 .92935550865482645193...  
  e1/ k  1
k!
k 1
k 1


.929398924900228268914... 


3  log 3  1 

0
2
 3
9
2  dx

 log x log 1  x
GR 4.222.3

k2

  2
1 .929518610520789484999... 
2
3
4
k 1 ( k  1 / 9 )

(1) k
 1 4 1
.92958455909241192604...  2 F1  1, , ,     k

 3 3 3
k  0 3 ( 3k  1)

.93002785739908278355... 
.930191367102632858668... 
263


csc  15
2310 2 15
e



.93052667763586197073... 
 (k )
 k !( k  1)!

k 2
.93122985945271217726... 
.93137635638313358185... 
6 .93147180559945309417... 
.9314760947318097375...



 1  1  1
  
I1  2
k  k  k 
k 1

 
1
log 2  3 
2

|  ( k )|

2k
k 1
2 arcsinh
10 log 2 
 (3) 
4
360


k Hk
 ( k  1)
k 1
3

1
(1) k  2k 
 
 k
2 k  0 8 (2k  1)  k 
1 .9316012105732472119...

.93194870235104284677... 
e  ee
2
2 

  3  4
32 2
cosh k
k
k  0 k !e


log x
dx
( x 4  1) 2
0
 
(1) k
.93203042415025124362...  
k  0 9k  1

log k
 3
3 .932239737431101510706...        3/ 2 
 2
k 1 k

4
1 


.9326813147863510178... 
  1 

2 sinh
2
16k 4 

k 1 

.932831259045789401392... 
1 .932910130264518932837... 
2
(1) k 1
dx
 
 2 log 2   3
   Li2 ( x 2 ) 2 
2
12
x
k 1 2k  k
0

2
37
  (3) 

16
12
1

 x Li (  x)
2
2
dx
0
(1) k

2k 
k  0 ( 2 k )! e

k7
2383 .93316355858267141097...  877 e  
k 1 k !

14
( 1) k
.93333333333333333333 =
 
k 
15
k  0 14
.93309207559820856354... 
1
cos 
e

AS 4.3.66, LY 6.110


1  ( 3)  1 
log 3 x dx
( 3)  3  
5 .9336673104466320167... 
          2
 4
 4 1 x  1
1256 
 1
1 .93388441384851971997... 
 1

(1) k 1 e k
e
.93401196415468746292...  1  e  1  cosh e  sinh e  

k!
k 1
e

 
=
e
x
dx
0
151 .93420920380567881895... 

.93478807021696950745... 
.93480220054467930941... 

413
( 1) k k 10

 
e
k!
k 0

2 7  4
k (k  12)
    

1
1
  6  3
k 1 ( k  3 )( k  6 )
2
2


 
J1061

(2k )!!
 3
  (1)   
2
 2
k 1 ( 2 k  1)!!k ( 2k  1)
 4
1
= 
2 
k 1 ( k  1 / 2)
( 1) k ( k  1) ( k )


2k 2
k 2


4
 (2 k  1)
k 1
2

1 .934802200544679309417... 
2 .934802200544679309417... 
4 .934802200544679309417... 

 
=
2
2

x yz
dx dy dz 
1  xyz
1 1 1
 3 
0 0 0

2

2 
2
0
2
x sin 2 x dx
1  cos x
 1
  (1)     i  log i 
 2
2

 1


 2,  
1
 2
2)


4k
(k  1)!(k  1)!4k
 
 

(2k )!
k 1  2k  2
k  21
 k
k 
1
 (k 
3 (2) 
k 0


 
=
(2k )!!
 (2k  1)!!k
k 1
2
2
4k 2  k  1
=  k  ( k )   ( k  1)   2
2 
k 2
k  2 k ( k  1)
2
= 2 arcsin 1

log 3 x
dx
= 
4
(

1
)
x
0


 

 


2

=
1 x
dx
 log  1  x  x(1  x
2
2
0

=
.93534106617011942477... 
 1  tan  dx
x
 log  1  tan x 
2
0
8 .93480220054467930941... 
)
GR 4.323.3
2
 1
 4   (1)   
 2
2
4

 ( 1) k
7
( 1) k 
  4
 1  
4 
729
( 3k  1) 4 
k  1  ( 3k  1)
5
.93548928378863903321... 
5 5 
2 2
2i
2 i
.93594294416994206293...   Li3 e  Li3 e
 
 0   

1
1 
sin      1 

2
5
25k 2 
1 / 5  k 1 
 


(1) k
.93615021799926654078...  log 2 

 
3 2
k  0 ( 2 k  1)( 3k  1)



1
.93615800314842911627...    1 

 ( k ) ( k  1) 
k 2 
.936342294367604449...
sin1


1   2

5 .936784413887151464...

GR 4.297.5
e
1
3

k 2
x dx

 e x
x

 1  

1 

k2 
2
.93709560427462468743... 

8
 /2
(1  2 log 2)  
 log(sin x) cos
2
x dx 
GR 4.384.10
0
1

 
=
  1  x 2 log x dx 
GR 4.241.9
0


 
=
 xe
x
1  e 2 x dx 
GR 3.431.2
0
7 .9372539331937717715...
63  3 7

1 .93738770119278726439... 
3   2 log 2
 

2
2
6
2


k2
 ( k )  1 

k 1 k  1
Adamchick-Srivastava 2.23
15
(1)
 
k
16
k  0 15

(1) k 1

.9375000000000000000000 =
.93750000000013113727... 

kk
k 1
.93754825431584375370... 
=
.93755080503059465056... 
=
k

log k
2
k 1 k


1
log n k

 k3
n  1 ( n  1)! k  1

( 1) k 1
   (2 ) 



kk
k 1
3
1 .93789229251873876097... 
2


log 2 x dx
  2

( x  1) 2
16
0
i Li3 (i )  Li3 (i ) 
1
=
k
log 2 x dx
0 x 2  1 
Berndt 8.23.8

x 2 dx
0 e x  e 2 x

log 2 x dx
1 x 2  1
GR 4.261.6

=
log 2 x dx
 4
x 1
0
 /4
=
 log tan x dx
2
GR 4.227.7
0
1 1

 
=
log( x 2 y 2 )
dx dy
2 2
1

x
y
0 0
 


  k H 1
1
15   2  3i  2 (i )   2 (i ) 
24

(1) k
.93809428703288482665...  
k  0 10k  1
.93795458450141751816... 

k
k 2
4
.93836264953979373002... 
.93846980724081290423... 
1
1
 4 sin  8 
2
2

( 1) k
 1
J1    
k
2
 2
k  0 ( k !) 16
8 cos
(1) k 1 ( 2 k )
.93852171797024950374...  

( k  1)!
k 1

k 2 Hk( 3)
6 .938535628628181848...
 
2k
k 1

( 1) k

k
k  0 ( 2 k )! 4 ( k  1)

k
k 1

1
2 1/ k 2
e

2 .93855532107125917158... 
  ( k )  1 ( k )  1 
1
k 2

2
( 1) k 1
1 .93872924635600957846...   2 csc

2   2 2
6
2
k  1 k ( k  1 / 2)

coth 
2
5 5
5  5 25 log 5 5
2
1
1 .93896450902302799392... 

log


4
4
2
5 5
5



5
1
 (k )
 
= 
 
  k 2 
k  1 k (5k  1)
k  0 ( k  1)( k  4 / 5)
k 2 5
.93892130408682951018... 




4 .939030025676363443...

3 ( 2) 
( 4)

2

4
256
2
23040
3
  where a( k ) is the nearest integer to k .
1
1 .939375420766708953077... 
 Li2 (3) 


1
 a(k )
k 1
4

AMM 101, 6, p. 579
log x
 x  1 / 3 dx
0
.93958414182726727804... 

3
33


H
1
2 2
log
  2kk 
2
2 2
k 1 2

kH
1
2 2
4 .9395976608404063362...  3  log 2 
log
  2kk 1
2
2
2 2
k 1
1 .9395976608404063362...

log 2 
 (k ) ( p )  1
.93982139966885912103... 
p k  kth prime
k
.9399623239132722494...

2 2  (6)
1  p 2

 
21
 (4) p prime 1  p  2  p  4
.94002568113...

g4 
J310
 (k )

.94015085153961392796... 
 (k  1)k !
k 2
1 .94028213339253981177... 

 ( 2 k  1)
k 1
k!

115

.94084154990319328756... 
384
e1/ k  1
 
k
k 1


sin5 x
0 x5 
GR 3.827.11

1 .94112549695428117124... 
2
24 2  32 
1  2 k  2

k 
 k 
8
k 0
16
(1)
 
k
17
k  0 16

9
(1) k
.94226428525106763631...  8 log   k
8
k  0 8 ( k  1)

.9411764705882352
4 .9428090415820633659...
=

.94286923678411146019... 
=
.942989417989417

k
=
4
2 2

3
2

CFG D4

sin k
1
 
  3 
6
4 12
k 1 k
i
Li  e i   Li3  ei 
2 3

7129
1
 
7560
k  1 k ( 3  k / 3)

Davis 3.32

HypPFQ [{1,1,1,1},{2, 2, 2, 2},1] 

( 1) k
= 
4
k  0 k !( k  1)
.94308256800936130684... 

 
.94318959229937991745... 
5  2

cot  2 
4
4


 
=

k
k 2
2
1

2
 2  (2k )  1
k 1
k 1
1 .943596436820759205057... 

 (2) (3)

 (6)

 (k )
 k   (k ) 
k 1


1  p 6


 
=

2
3 
p prime  (1  p )(1  p ) 


(1) k
5 .94366218996591215818...  3  2 6 csch
  2

6
k 0 k  1 / 6


1
1 
 

p ( p  1) 
p prime 
.94409328404076973180... 
 (3)
4
1


0
 arcsin 2 x  2 arcsin 3 x
x
dx
 /2
=
x

2
log(2 cos x) dx
0
Latter result due to Peter J.C. Moses and Seiichi Kirikami (A193712)
( 1) k  2 k 
4 58   k
 
k  0 16 ( k  1)  k 
80  4 5

17
(1) k
 
k
18
k  0 17
1
(1) k
cos  

3
( 2 k )! 9 k

.9442719099991587856...

8 .9442719099991587856...

.94444444444444444444
=
.94495694631473766439... 
 1
.9451956893177937492...   2 Li3    
 2

 
log 2 x
= 
dx 
2x2  x
1
AS 4.3.66, LY 6.110
( 1) k

k
3 
k  0 2 ( k  1)


x2
0 2e x  1 dx
1
=
log 2 x
0 x  2 dx
 0 (k )

.94529965343349825190... 
 (k  1)!
k 1
8 
 5
    
15
 2


 (2k )
  2

1 .9455890776221095451...  

 I 0    1

2
k

k  1 ( k !)
k 1 
.94530872048294188123... 
1 .94591014905531330511... 
 6
log 7  Li1  
 7
1
.94608307036718301494... 
si (1) 
sin x
dx 
x
0

( 1) k
= 

k  0 ( 2 k  1)!( 2 k  1)


 


 
=
AS 5.2.14
 1  dx

x
 sin x 
1
1
=
  log x cos x dx
0

4 .94616381210038444055... 
.946190799031120722026... 
3 e 
pf ( k ) k

2k
k 0

log 6  2  2 

 (1)k
k 2
2k
 (k )  1
k

1 .9467218571726023989...

k
 sin 2
k 1

.946738452432832464562... 
 (k )

k!
k 1
5 .94678123535278519168... 
 5 tan
 

2 5


k
k 0
2
1
 k 1/ 5

sin 3 k
1 cos 1
cos 3
 3e sin(sin1)  e sin(sin 3)   k ! 
4
k 1
i
i
i
 3i
3i
3ee  3ee  ee  ee
8

(1) k 1
7 4
7 ( 4 )
  (4) 
 

720
8
k4
k 1
  1  i 
  (1  i )  
  cot  (2 )1/ 4  coth 1/ 4   
3/ 4 
 2

2  4 
.94684639467237257934... 

k

=
.94703282949724591758... 
.947123885654706166761... 

J306

=
k2

4
k 1 k  2

.94716928635947485958... 
tanh
11
18
.947368421052631578947... 

19
2
(1) k

k
k  0 18


k
k 1
2
1
k3


2
16
1 .94752218030078159758... 
 11
2
  2  8 log 2  8 log 2 2 


 
=
e
 x2
log 2 x dx
GR 4.335.2
0
.947786324594343627...
 sin

.94805944896851993568... 
1





 1  
k 1

1 

k2 
3
1 
 1
1 2  



 8k  7 8k  1
8
J78, J264
k 1

Q( k )
k 1 k !
 ( 2 )   ( 3)
.94899699000260690729... 

3
1
4 .949100893673262820934... 
 ( 3)  1
1 .94838823193110849310... 
.94939747687277389162... 

arctan(sinh 1csc 1) 

 (1)
k  

 


k 1
 1 
arctan

 k  1
[Ramanujan] Berndt Ch. 2, Eq. 11.4

k

.9494332401401025766...   k
k 1 3  1
.949481711114981524546... 
.94949832897257497482... 
.9497031262940093953...

 2

 
=
9 .9498743710661995473...

3k

( 1) k

k
1
k  0 k !( k  2 ) 4

 ( 1) k
( 1) k 

 1  
2 
(6k  1) 2 
6 3
k  1  (6 k  1)

0


k 1

2


1 ( k )


6
2 sin 1


log x
dx 
x6  1
99 
J329
.95000000000000000000
=
.95023960511664325898... 
.95038404818064742915... 
.95075128294937996421... 
.95105651629515357212... 
.9511130618309...

1 .95136312812584743609... 
.9513679876336687216...

.95151771341641504187... 
(1) k

k
k  0 19

19

20
2
1 5
 3 log

6
2
2

 
.9520569031595942854...

3 1
1
   2

2 3
6
k 1 k  3 / 2

8
 ( 2k )
 
7 (3)
k3
k 1

1
2
10  20  sin
 cos
4
5
10

(k )
( 1) k 1 4 

k
k 1

(1) k 1 8k
2 sin 2 2  
( 2 k )!
k 1
2

15  e
2k
 
8
k  0 k !( k  6)

1  (1 )  1 
(1) k
(1 )  2  
         
2 
 6
 3 
36 
k  0 ( 3k  1)

log x
=  
dx 
1  x3
0

Berndt Ch. 9
coth 
1

(1) k ( k !) 2

2 
k  0 ( 2 k )!( 2 k  1)

 (3) 
1

4
x log x
1 1  x 3 


  k
k 2
2

e
0
x
x dx
 e 2 x
1
1 
 5

 k k  k3

=
  ( k )   (2 k  1)
k 2
.95217131705368432233... 
=
.95247416912953901173... 
2 .95249244201255975651... 
 cos x log(sin x )dx
0
20
(1)
 
k
21
k  0 20
1  1
1
   , 2,  
4  2
2

.952380952380952380
 /4
2  log 2
 
2 2
k
( e  1) 2 



j 1 k 1
(1) k

k
2
k  0 2 ( 2k  1)
1
j!k !

e
1
3
  1  tanh  
e 1
2
2
3
.95257412682243321912... 
.9527361323650899684...

3

2 e


1
 1 t
0
2
cos
t
dt
2
 (1) e
k
3k
J944

5 .952777785411260053338... 

 

11 4
11 (4)
1
x


  x 2 log 2  2 cos  dx
180
2
 0
2


Borwein & Borwein, Proc. AMS 123, 4 (1995) 1191-1198

.953530125551415563988... 
  (k )

   (k  4)  1



551
k5
.95353532563643122861... 
 101e  

2
k  0 ( k  3)!
n


 (k )
k (k )
 
.954085357876006213145...  
, where  ( n)    ( k )
k  2 k!
k 1 ( k  1)!
k 1
k 2

4 .95423435600189016338... 
.95460452143223173482... 
3 .954608700594592236394... 
.95477125244221922768... 
Ei ( 2 )    log 2 
1  ( 4)   (5)
 2 (3)
3

2k


k 1 k ! k
AS 5.1.10
 2 (2) (3)
3
log 3  log 2 
2
1

1
 log1  2 x  dx
0
2
7 .954926521012845274513... 
2 I 0 (1) 
e
sin x
dx 
0
.95492965855137201461... 
=

 
=
2 .95555688550730281453... 
.95623017736969571405... 

 
=
2 .95657573850025396056... 

1 

  1 

 k 1  36k 2 

  
cos


 6  2k 
k 1
3
GR 1.431
GR 1.439.1
 0 


 1 / 6

 2k   ( k )  1
 

k!
k 2  k 

7
3 ( 3)

  k 3  ( 2 k  1)  1 
128
4
k 1
4
2

k ( k  4 k  1)

( k 2  1) 4
k 2

1587
k6

293e 
 
2
k  0 ( k  3)!
.956786081736227750226... 
1  cosh   sinh  
.95698384815740185727... 
5

162

e
x
dx
0
3
sin k / 3

k3
k 1


GR 1.443.5
G
.95706091455937067489... 


6 .9572873234262029614...

 x  ( x)  1 dx
2
2

.957536385054047800178... 

 log (s n)
s  2 n 1
.957657554360285763750... 
1 .95791983595628290328... 
2980 .95798704172827474359... 
.958168713149316111695... 
1
 cot 2 
2
 arcsinh 1
2

1
2
k 1

k
tan
1
2
k 1

Berndt ch. 31



2 log 1  2


log( x 2  1)
0 x 2  2
8
e
2
2
 

1
1
 (1  i )   (1  i )   ( 2 ) (1  i )   ( 2 ) (1  i )

2
8
5i (1)
 (1  i )   (1) (1  i )  2 ( 3) 

8


 
=
 ( 1) k  ( k )   (2k  1) 
k
2
k 2

.95833608906520793715... 
.95835813283300701621... 
.95838045456309456205... 
( 1) k 1
2
k 1 ( k )!

1
1
(1) k
cos
cosh
 
2
2
k  0 ( 4 k )!
1  ( 2 )  5


 3
 1
     ( 2 )    2 ( 2 )    2 3 cot csc2 
 8
 8
 8
1024 
8
8

=
 (1) k
(1) k 

1  
3 
( 4 k  1) 3 
k  1  ( 4 k  1)
=
(1) ( k 1) / 2 


3
k  0 ( 2k  1)


.95853147061909644370... 
1   3   1
 cos
    
3   2
3  e
(1) k

k  0 ( 3k  1)!


1  tanh( / 2)
e / 2
  /2

( 1) k e  k 

2
e  e / 2
k 0
 1  1  3 
2 .95867511918863889231...       
 4
 4


1
( 1) k
1 

.9588510772084060005...  2 sin  

1  2 2  

k
2 k  0 (2 k  1)!4
4 k 
k 1 
.95857616783363717319... 
J944
GR 1.431
=
 0 


 1 / 2 
(1) k 1

2
k  1 ( 2 k  1)! k

.95924696800225890378... 
4  ci (1) 
.95951737566747185975... 
1
e


2 sinh(1 / 2)
e 1
.95974233961488293932... 
.95985740794367609075... 
2 2 / 3

3 3


e
 ( k  1/ 2 )

J942
k 0
x dx
3
2
x
0
1
1
1
 (  1) cos  log2  2 cos1sin  
2
2
2

1
 k sin
k 1
2k  1
2

.9599364907945658556...

  (k )  (k  1)  1
k 1
k ( 2 ( k )  1)


k!
k 2

16
1
( 1) k 1 ( k !) 2
.96031092683032321590... 
arcsinh  

k
4
17
k  0 ( 2 k  1)! 4

4
1
(1) k

4

2

.9603271579367892680... 

erf

k
2
e1/ 4
k  0 ( k  1)! 4 ( 2 k  1)

1 .96029978618557568714... 
.96033740390091314086... 
1 .9604066165109950105...

32 2

15

2


1 
2

  1 16k
k 2

log 2 log 2 2
 


48 8
4
4
 /2
 log sin x 
2
0

.96090602783640284933... 
.960984475257857133608... 
.96120393268995345712... 
1 .96123664263055641973... 
.961533506612144893495... 
Hk

2 log 2 2  
k  1 k ( 2 k  1)

 (k )
(1) k 1 k

2 1
k 1

1
(1) k
2 2 arctan
  k
2 2
k  0 8 ( 2 k  1)
3 ( 3)   ( 2 )
8  8 
11 2
 Li2  e 4i   Li2  e  4i 
6
4 ( 3)
5
2


 ( 2 k  1)
ek
2 .96179751544137251057...  
  3
k 1 ( k  1)!
k 1 k
4
1 
(1) k

.96201623074638544179...  4  log  arctan    k
5
2  k  0 4 (2k  1)(k  1)

.96164552252767542832... 
cos2 x dx

1
1 5
(1) k  2 k 
 2 log
  k
 
2
2
k  0 16 ( 2 k  1)  k 

1
.96257889944434482607...  log 7  2 

arccot 3 3 
2 3
3

1 
(1) k  1
 
= 1   3k 1 


 6k  1 6k  1
k 1 3
.962423650119206895...


2 arcsinh


1 .962858173209645782869... 
1
L
k 1
.96307224485663007367... 
2   (5)
.96325656175755909737... 
5 (1)   (1)  (1)   (1) 
1/ 5



k
 
=

3/ 5
4 /5
1

1
5

  1  k
k 2
5

8

1
 32k  6
k 1
 (1 / 2)
 1
.963510026021423479441...    2 log 2  1      
1
 2
 (1 / 2)
 3
 
=      1 
 2
1 .96349540849362077404... 

2/5
 32k
2

.96400656328619110796... 
 (1  log 2 ) 
log(1  x 2 )
0 x 2 (1  x 2 ) dx
e2  e 2

e2  e  2
.9640275800758168839...

tanh 2 
3 .9640474536922083937...

(i  1)  ( i )  (i  1)  (i ) 

.9641198407218830887...

.96438734042926245913... 
.96534649609163603621... 
.96558322350755543793... 
.9656623982056352867...

2
3 ( 3)
2 
 4 log 2 

6
4

 (k )
1
  5
 (5)
k 1 k

 (10)
 (k )
  5 
 (5)
k 1 k

1

k
k 0 2  2

 (2 k )
  2 k 1
1
k 1 2
( 1) k 1

4
3
k 1 2 k  k

HW Thm. 300
.9659258262890682867...

.96595219711110745375... 

 6  5 3 
108
3

.96610514647531072707...  erf
2
4 .96624195886232883972... 
1 .9663693785883453904...

5
1
 sin

12
6 2
2
1 3 
4

AS 4.3.46

log x
dx
( x 6  1) 2
0
 
(1) k 32 k 1
 k !22 k 1 (2k  1)

2
 ( 2 )   ( 3)   ( 4 )   (5)
  i 
1  1 1
 i 
   , 3 ,   i 2  Li3  
  
  Li3 
4  2 2
2
 2 
 


 
.9667107481003567015...
=
log 2 x
1 x 2  1/ 2 dx

1
 cosh1 sin1  cos1sinh1 
2

4 .96672281024822179388... 
1
 cos x cosh x dx
0
1

  1 k !! 
k 1
.96676638530855214796... 

 
=
1 .96722050079850408134... 
.96740110027233965471... 
.96761269589907614401... 
.9681192775644117...

.96854609799918165608... 
7

sin


1 

  1 

7
49k 2 
k 1 
 0 


 1 / 7
 

3
8
1 1 1 1
 4G  2 log 2 
0 0 0 0


5
1  ( 1)  1 
 5 
      (1 )    
 8
 8 
64 
log x
dx 
1  x4
0


=
3 .96874800690391485217... 

w x y z
dw dx dy dz
2 2 2 2
y z
    1 w x
3
  (k )   (k  1) 
  (1) k k 2 
 1 
4
2
2


k 2

 ( 4 k  3)

4k 3
1
k 1 2
2
1
=
GR 1.431
x log x
1 x 3  x 1 dx 
(1) k

2
k  0 ( 4 k  1)


x 2 log x
1 1  x 4 dx

x
1
log x
dx
 x 2
2

2
5 1 5
 25  2 5 log 2  arcsinh 2   5  5 log 1  5   51  5 arccsch 2

=
4 .96888875288688326456... 
.96891242171064478414... 
.96894614625936938048... 
=
=
Fk Hk
k
k 1 2

 2 (k )

k  2 ( k  2 )!
1
cos  Re (1)1 / 12 
4


3
(1) k
  ( 3)  
3
32
k  0 ( 2 k  1)
i
Im Li3 (i )    Li3 (i )  Li3 (i )
2

sin k / 2

k3
k 1

1 1 1
=

(1) k

k
k  0 ( 2k )!16

AS 23.2.21
dx dy dz
2 2 2
y z
  1 x
0 0 0

5 .96912563469688101600... 
2k

k  1 k ! ( 2 k  1)

.969246344554363083157... 



   (4k )  1 
k 2


2
csc

 1
12
2 2
2

( 1) k 1
 
= 
4
2
k 1 2 k  k
4 5
 g5 
.96944058923515320145... 
729 3
.96936462317800478923... 

  (k )
Titchmarsh 14.32.3
2 .9706864235520193362...

9 .9709255847316963903...

.97116506982589819634... 
2

 2   2
G 1/ 3
43867
 B9
798
2
2
cos x
sin x dx
 2

.97201214975728492545...  2 
1  
dx   
 3  0 2  cos x
0 2  sin x
54 .9711779448621553884
=

3 .972022361698548727395... 
 k  (k ) (k  1)  1
k 2
J310

k 4 Hk
379 .97207708399179641258...  276  150 log 2  
k
k 1 2

15 (5)
(1) k 1
.97211977044690930594... 
  (5)  
   (1, 5, 0) 
16
k5
k 1

15 (5)
(1) k
.97211977044690930594... 
 
5 
16
k  0 ( k  1)

2 2
log 2

log(1  2 ) 
3
2
3
3

1
 
= 
k  0 ( k  1)( 2 k  1)( 4 k  1)

1
(1) k
.97245278724951431491... 

6 sin
  k
6
k  0 6 ( 2 k  1)!

1 

=  1 

6 2 k 2 
k 1 
log 7
.97295507452765665255... 
2


(1) k
(1) k 

= 1    3k  1
 3k
( 6k  1) 3 ( 6k  1) 
k 1  3
.97218069963151458411... 

 2
AS 23.2.19


K ex. 108

1 .97298485638739759731... 
 H  (k )  1
k 2
1 .97318197252508458260... 
k 1
5 3
13 (3)

8
81 3

=
=
=
.97336024835078271547... 
2 .97356555791412288969... 
1  (2)  2 
x 2 dx
( 2)  1  
          x
 3
 6 
216 
e  e 2 x
0
 ( 3) 2
1/ 3
 (1)1/ 3 Li3   (1) 2 / 3   (1) 2 / 3 Li3  (1
2
3


1
2
log x
log 2 x
x log 2 x
0 x 3  1 dx  1 x 2  x 1 dx  1 x 3  1 dx

1
 1
 2  2
   
2 / 3    
 3
 3  3
(2 )
4 ( 2 )  3 ( 3)
2 2  75

1 .973733516712056609118... 
48

S2 ( 2 k , k )
.97402386878476711062...  
( 2 k )!
k 1
.97407698418010668087... 
log(1  e )

kH k H k
 (k  1)(k  2)(k  3)
k 1

.974428952452842215934... 
3e  3


log 2 x
0 ( x  1)( x  e) dx
  i
 i 
i  Li2     Li2    
 2 
  2

1  1
1
(1) k
   , 2,    k
 
=

4  4
2
k  0 4 ( 2k  1)( 2k  1)


8
2
  
sin 
8  2 2   cos k  3 
.97449535840443264512... 


8
2 
k 1

(1) k
1
.97499098879872209672... 
  2 log 1  2  
2
k 0 k  3 / 4
.9744447165890447142...

 2 1


=
 (1  x
0
1 .97532397706265487214... 



2
GR 1.439.1
dx
) cosh x / 4
GR 3.527.10
3 4  2 log 2  log 2 2



64
16
16


0
log 2 x
dx
x2  4 x2
  
sin    Im(i )  
 2
2
.97536797208363138516... 

1 .975416977098902409461... 
1 .97553189198547105414... 
25 .97575760906731659638... 
1 .97630906368989922434... 
3 2 log 2 21 (3)


4
8
2 sin 2 
k 1
(1) 16
(2k )!
k 1


4 4
 24 ( 4 ) 
15
 /2

0
x3
dx
sin 2 x
k

x4
0 ex  e x  2
I 0 (1)  L0 (1) 
1

 
=
 exp(sin  x) dx
0

1
11 4
3
Hk
 ( 4)   2 ( 2)  
2
2
360
2
k 1 ( k  1)


1
(1) k  1
.97645773647066825248...   sin k  
2 k 1
2
 1)
k 1
k  1 ( 2 k  1)!( 2
2


1

1

cos    1  2
.976525256762151736913... 
2
2
 1 2
 (2 k  1) 
k 1 
2
2 .9763888927056300267...

1  (1)  1 
 5 
     (1)    
36 
 6
 6 
2

  ( 3)
 (k )
1 .9773043502972961182...   ( 2 ) ( 3) 
 13 
6
k
k 1
.976628016120607871084... 
18 MI 4, p. 15
h2 
J314


k 1
 1 ( k )
k2

HW Thm. 290
 2 log 2
7 (3)
.977354628657892959404... 
  (2) 

2
3

k 1  ( k )
.97745178 ...
  ( 1)
k3
k 1

3 .97746326050642263726... 
 I 0 (1) 

 k  12 
k 1
k2


 ecos x dx 
e
0
0
cos 2 x
dx

 2 2
log 2 2 
2 .9775435572485657664...    1   

 log 2   log 2 

2 24
2 


=
.977741067446923797632... 
4 .9777855916963031499...

log 2 x sin 2 x
0 x 2 dx
1
2   1   
 4

2k  2 

 1  2k  3  
k 0

3
k  2k 
e I 0 (1)  I1 (1)   1F1 ( ,2,2)   k  
2
k  0 k!2  k 
35
109 .9778714378213816731...  55 

2

2k k 3


k  0  2k 
 
k

7G 3
(1) k 1 k 5
   2
3
4
8
k 1 ( k  1 / 4 )
1  ( 1)  1 
 3 
.97797089479890401141... 
      ( 1)    
 10
 5 
100 
1 .977939789810133276346... 


 
=
x log x
1 x3  x  2 dx 

2
.97829490163210534585... 
G 1/ 4
1 .9781119906559451108...


2
6

1
log x
dx 
5
1
x
0
2
log xdx

ex
0

(1) k

2 
k  0 (5k  1)

GR 4.355.1
40320 (9)    ( 8) (1)
 3
.97846939293030610374...  Li2   
 4
40400 .97839874763488532782... 
.97895991797814145164... 

4 2
 /2

 
=

0
log (3  2
x dx
sin x  cos x


2 )  i   2arc cot(1  2  2arc cot(1  2 ) 
.97917286680253558830... 
.97929648814722060913... 
.97933950487701827107... 
.97987958188123309882... 
.98025814346854719171... 

( 1) k
1
1
cos 3/ 4 cosh 3/ 4  
k
2
2
k  0 ( 4 k )! 2

1
(1) k
2 2 sin
 
k
2 2
k  0 ( 2 k  1)!8

1
( 1) k
16 sin2  
k
4
k  0 ( 2 k  1)! 4 ( k  1)
 log
1 3

2
1

log(1  x 2 )
dx
1  x2

 2k  (1) k
2 log 2     k
k  0  k  32 ( 2k  1)
GR 4.295.38
0

k3
 (k  1)  1

k 1 k  1

1
( 1) k
.98158409038845673252...  3sin   k

3 k  0 9 (2 k  1)!

1 

=  1 

9 2 k 2 
k 1 
2 .981329471220721431406... 
.98174770424681038702... 
  2 (3) 
5

16

.98185090468537672707... 

k 1
.98187215105020335672... 
=
1 .98195959951647446311... 

 (2k )
k !2
k
x 5/ 2
dx
1 x


 (2k )
GR 3.226.2
 (2k )!!   e
k 1

.98265693801555086029... 
1/ 2 k 2

6
5

1
k 1
i
Li2   (1) 3/ 4   Li2  (1)1/ 4 
2

sin  k / 4

k2
k 1
2
3
3


csc
5
5
5 5
1  tanh 2
e2
.98201379003790844197... 
 2

2
e  e 2

19 e1/ 3
k3
.982097632467988927553... 
 
k
27
k  0 k !3
1 .9823463240711475715...




dx
 1 x
5/ 3
0

 (1)
k
e 4 k 
J944
k 0

 2 e x 2
22 2 log 2
2 1  e2 x  dx

  
  2 
 GR 3.438.2
9
3
3
x x
x3  x
0
2   ( 6)

1  cosh  3
1
i 3 
2 
 
 

.98268427774219251832... 
2
2 sin
12
6
2 
2


1
6

 1  k
k 2
.982686076702760592745... 
.98295259226458041980... 
.983118479820648366...
 
945




(k )
5 .98358502084677797943... 
.98361025039925204067... 
.98372133768990415044... 
1
log 2 x
0 x 2  2 
i   i 
 i 
 Li3  
 
  Li3 
 2 
2 
2
691 6

675675
1
6
p prime 1  p

 ( 2 )   ( 3)   ( 4 )   (5)   ( 6) 

( 1) k 1
 3

 2G  log 2  
3
2 16
k 1 k ( 2 k  1)

1  (1 )  1 
(1) k
 7 
      ( 1)     
2
 12 
 12  
144 
k  0 ( 6k  1)

=
k 1
1  1
1
   , 3,  
8  2
2
=
.983194483680076021738... 
6
1

 (6)
(1) k

k  0  2k 
 (2k  1)3
k 

k6

1
1 

  sin
  1  4 2 

 k 
k 1 
.98318468929417269520... 

 1 1   3 3 3  1 
HypPFQ  , ,1,1,  , , ,   
 2 2   2 2 2  4 
x log x
dx
3
 x 3
x
1
3k 2  3k  1
2  4 ( 2 )  2 ( 3)  2
3
k  2 k ( k  1)

10 .98385007371209431669... 

=
 (k
k 2
.98419916936149897912... 
.9843262438190419103...
.9845422648017221585...


2
 k ) ( k )  1
403 6
 6
393660
23
  ( 6)   ( 5)
63
2
 (3)
 2 log 2 2 

6
4
.98481748453515847115... 
( 1) k

k 4
k 1 k
.98517143100941603869... 
2
.98542064692776706919... 
 1
log    
 3
J312
(1  x ) log 2 (1  x )
dx
0
x2
1



2  1 21/ 4 , from elliptic integrals

16
(2 k )



4

k4
15 ( 4 )
k 1

(1) k 1
31 6
31 ( 6)
.9855510912974351041... 
  ( 6) 
 
30240
32
k6
k 1

3
( 1) k


4 .98580192112184410715...  4 2  log sin
 log sin   
1
3

8
8
k  0 ( k  4 )( k  4 )
.9855342964496961782...

1 .98610304049995312993... 
96

1  ( 2 )  5
( 2 )  1 
        
 8 


512 
8
J306
1
log 2 x
0 x 4  1 dx 


 
=
.98621483608613337832... 
6 .98632012366331278404... 
.9865428606939705039...
log 2 x
1 x 2  x 4 dx
( 1) k

k
k  0 ( 2 k  1)! 4 ( 2 k  1)

5e2  9
2k k 2
 
4
k  0 k !( k  3)
 1
2 si   
 2


11 4
 1
768 2
=
(1) ( k 1) / 2 

4
k  0 ( 2k  1)
 ( 1) k
( 1) k 



4 
(4 k  1) 4 
k  1  ( 4 k  1)

J327, J344

.98659098626254229130... 
=
12 .98673982624599908550... 
.98696044010893586188... 
Prud. 5.1.4.4
5 3
13 (3)
1  ( 2)  2 
 1 
     ( 2 )   


2
432 
162 3
 3
 6 

(1) k

3
k  0 ( 3k  1)

28402
(1) k 1 k 10

2187
2k
k 1
2
10
1856 380
4
8 .98747968146102986535... 

log 
243
81
3
.98776594599273552707... 
.98787218579881572198... 
=
6 .98787880453365829819... 

k 4 Hk

k
k 1 4
sin 2 
4
7 4 15


csc  4 15  csch  4 15
105
420

(1) k

4
k  2 k  15
 
2 4
6 
15

( x  1) log 3 x
0 1  x dx
1


.98829948773456306974... 
.98861592946536921938... 
2


18 3  1
3 2

3
2

 
=
0
log x
dx 
x12  1



 cos  3  2
k 1

 

1 

3  1   1 

144 k 2 
k 1 

=

k 2
GR 1.431



 0 


 1 / 12
2 2 k 1
sin 1 sinh 1   ( 1)

( 4 k  2 )!
k 0

1 

=   1  4 4 
 k 
k 1 

.98889770576286509638... 

 
k
GR 1.413.1
1  ( 3)  1 
 3 
      ( 3)    
 4
 4 
1536 
i
=   Li4 (i )  Li4 ( i ) 
2


(1) k
sin k / 2
= 
4  
k4
k  0 ( 2 k  1)
k 1
.98894455174110533611... 

 

 
1 .98905357643767967767... 
.9890559953279725554...


3
log 2 x dx

  6
x 1
9 3
0
2

2
log 2 (2 x)dx

 
2
12
e2 x
0

log 2 k
k2
k 1

1 .98928023429890102342... 
  ( 2 ) 
1 .989300641244953954543... 
 (3  i)   (3  i ) 

.98943827421728483862... 
k 1
.98958488339991993644... 
.98961583701809171839... 
=
 0 (k )
  2k 
4
 (2  i)   (2  i)
5

 
k

1
1
( 1) k
cos cosh  
k 
2
2
k  0 ( 4 k )! 4

1
( 1) k
4 sin  
k
4
k  0 ( 2 k  1)!16

1 

1 


16 2 k 2 
k 1 

.98986584618905381178... 
1 .98999249660044545727... 
.99004001943815994979... 
4 arcsinh
1

4
 2k 
( 1) k



k
k  0  k  64 ( 2 k  1)

1  cos 3  2 sin 2
3

2
( 1) k 1 9 k


(2 k )!
k 1

3
1  (2)  5 
 7
 1   (7  4 3 )
     ( 2 )    2 ( 2 )    
 12 
 12 
 12  
3456 
432
 ( 1) k
( 1) k 
1  

3 
(6k  1) 3 
k  1  ( 6k  1)

=
.990079321507338635294... 
1  2
7
 15  
 4  20  14 2   (1)     (1)   
256 
 16 
 16  

=
log x dx
2
 x 6
x
1
10
74
4 .990384604362124949925... 


3 81 3
78 .99044152834577217825... 
1
212

e

.99076316985337943267... 

k3


k 1  2k 
 
 k
(1) k k 10

k  0 ( k  1)!

 k  (k )  1
2

k 2
.99101186342305209843... 
21 .99114857512855266924... 
5 .991155403803739770668... 
1
 (i  1) ( i )  (i  1) (i )
4
7
1 
 7 
3
176  4 2   (3)     ( 3)   
4096 
8
 8 

4 .9914442354842448121...
=
log 3 x dx
1 x 2  x 2

6
 ( 3)
.99166639109270971002... 
.99171985583844431043... 

1 2 3 4  1 
HypPFQ  ,  , , , ,
 
 5 5 5 5  3125


 (k )
1
  7
 (7)
k 1 k

1
 (5k )!
k 0
7
6 .99193429822870080627... 
 (k )
k 2
2 .9919718574637504583...
.99213995900836...
2  3

3

7
Borwein-Devlin p. 35

1 .992173853105526785479... 
I
k 0
.99220085376959424562... 
.99223652952251116935... 
1 .992294767124987392926... 
1 .992315656154176128013... 
.99259381992283028267... 
.99293265189943576028... 
(1)

4 3
sin 2 k / 5
 
125
k3
k 1
56 7
 g7 
94815 3
e(e  1)
1

   ,  2, 0  
3
(e  1)
e

GR 1.443.5
J310

k2

k
k 1 e
5
 i Li5 (i )  Li5 (i ) 
768

63 ( 7 )
(1) k 1

  (7)  
k7
64
k 1

2

1
(1  e ) 

.99297974679457358056... 
k

 sin(tan x)
0
AS 23.2.19
dx

x
GR 3.881.2
 ( 3k  2 )
2 3k  2  1
1  ( 2 )  5
 1 
     ( 2 )   
 8
 8 
1024 
k 1
.99305152024997656497... 
2
 
 
.99331717348313360398... 
2  2 
.993317230688294041051... 

i 
i 

2  2  1    1    
 2   2 


.993703033793554457744... 
.9944020304296468447...

.99452678821883983884... 
1 .9947114020071633897...

.99532226501895273416... 
3
  Li2 e 2i  Li2 e  2i 
17 (4)
 3 (3) 
4

k
k 1
6 log 2  log 2 2  2 

 k (k
k 1
2
1

 1 / 4)
Hk Hk
2
(k  1)
2
6



k3

k
2
k  1 2 ( k  1)
3
18 3
5
2
 h3
J314
(1) k 2 2 k  1
erf 2  
 k  0 k !( 2 k  1)
2



1 .9954559575001380004...

dx
x
x
0
1 .99548129134760281506... 
.99551082651341250945... 
10 .99557428756427633462... 
21 .99557428756427633462... 
.995633856312967456342... 

2

 log x sin x
(  log 2 ) 
0
(91  45 5 ) 4
 1
18750
7
2
7
 11 
2
dx

x
GR 4.421.1
 ( 1) k
( 1) k 




4
(5k  1) 4 
k  1  (5k  1)


2k k 2

k 1  2k 
 
 k
1  ( 4 )  3
 5
 1
 7
     ( 4 )     ( 4 )     ( 4 )    
 8
 8
 8
 8
786432 

 
 (1) k
(1) k 
= 1   

5 
( 4 k  1)5 
k  1  ( 4 k  1)

 
=

(1) ( k 1) / 2 

5
k  0 ( 2k  1)


.99585422505141623107... 

Prud. 5.1.4.4
 2 (k )
2k
arctan k
.99591638044188511306...  

2k
k 1
sinh 
1/ 4
.99592331507778367120...  
sin ( 1) 3/ 4  

3 sin ( 1)
8
sinh 
 
=
cosh  2  cos  2 
16 3
k 2






1
8 

 1  k
k 2
8
7 .99601165442664514565... 
 ( k )
k 2

k3
1 .99601964118442342116...  142 e  384  
k  0 k !( k  4 )
8
5207
.99608116021237162307... 
 8 
49601160
15 .996101835651158622733... 
.99610656865...

32 238


3
81 3
g8 
J312

k4


k 1  2 k 
 
k
J310

(1) k
5 5
  (5)  
.99615782807708806401... 
5 
1536
k  0 ( 2 k  1)
127 8
127 (8)
.99623300185264789923... 
 (8) 

1209600
128
 
( 1) k 1


k8
k 1

J306
19 5

4096 2

 ( 1) k
( 1) k 
= 1   


5
(4 k  1)5 
k  1  ( 4 k  1)
.99624431360434498902.. 

AS 23.2.21, J345
.996272076220749944265... 
.99633113536305818711... 
8 .99646829575509185269... 
2 
tanh  
 log 3
tan i
e  e  
 
i
e  e
1/ 2
K (k )k
dk 
4 3/ 4 0
2 1
2
k
(k )
2
 3
x sin x dx
3
2 log 2  9 (3)  
1  cos x
0


GR 6.153
k 2 ( k )
2k
k 2

e(e 4  263  66e 2  26e  1)
k5
1

   , 5,0    k
119 .99782676761602472146... 
(e  1) 6
e
 k 1 e
2
2

3
11
7 ( 3)
16k  6k  1
.99784841149774479093... 

 log 2 
 
3 
8
2
4
k  2 2 k ( 2 k  1)

k 2 ( ( k )  1)

 
= 
2k
k 2

.997494864721368727032... 
 ( 1) k
9
8 .99802004725272736007... 
 (k )
k 2
.998043589097895...

9 
J312
3236
 g9 
55801305 3

 ( 1) k
23 4
( 1) k 
.9980715998379286873... 
 1  

4 
( 6k  1) 4 
1296 3
k  1  ( 6k  1)
9
.9980501956570772372...

255 (9)
.99809429754160533077...   (8) 

256

2 .998095932602046572547... 
Hk Hk
 k (2k  1)
k 1
( 1) k 1


k9
k 1
J310

AS 23.2.19
e 2
1  tanh 
.99813603811037497213... 

  (1) k e  2k 
2
2
1 e
.9983343660981139742...


3
1 
  
 128 2 64 
 3


1
  (8k  5)
3

J944

1

3
(8k  3) 

( k  1)( k  2 )( k  3) Hk
1016 512
4
17 .99851436155475931539... 

log  
81
27
3
4k
k 1
k 1


.99851515454428730477... 
 3  2 
.99857397195353054767... 
19 
 1
14
2 3 5 3
.9986013142694680689... 
9 2

 log1  x
2
0
2
2
4
1 
 dx
 2
 ( 1) k
( 1) k 



6 
(4 k  1) 6 
k  1  ( 4 k  1)

J327

1 
(1) k

 1 1 1 2 5
.99861111319878665371...  HypPFQ   ,  , , , , , 
   ( 6k )!
 6 3 2 3 6  46656 

k 0

1  ( 5)  1 
( 1) k
( 5)  3  
.99868522221843813544...   ( 6) 
 AS 23.2.21
         
 4
 4   k  0 (2 k  1) 6
491520 
.998777285931944...
 h4 
J314
33 .99884377071514942382... 
.999022588776486298...

.99903950759827156564... 
1 .99906754284631204591... 

5e4  1
4k k
 
8
k  0 k !( k  2 )
10
48983
 10
4591650240
73 10
511 (10)
  (10) 

6842880
512
6 2 
 4 sin 1sinh 1

4 1

(1) k 1


k 10
k 1

J306
4 

k4 
k 2

e e  e 2 1/ e  e 2  1
sinh k
 
1 .99916475865341931390... 
2e
k 1 ( k  1)!
7
1 .99919532501168660629... 

11

3 7  4 3
log 2 x dx
1 .99935982878411178897... 
   12
x 1
216
0
.99915501340010950975... 

.99953377142315602297... 
.9995545078905399095...


 1  
4

3 2

( 1) k
61 7
  ( 7)  
7 
184320
k  0 ( 2 k  1)
AS 23.2.21
.99955652539434499642... 
2
307 7 2
 1
1310720
11 5
 h5 
1944 3
305 5
.9997427569925535489...  2 
 1
93312
.999735607648751739...
.9997539139218932560...
 ( 1) k
( 1) k 




7
( 4 k  1) 7 
k  1  ( 4 k  1)




 
=

 
=

J314
 ( 1) k
( 1) k 




5
(6k  1) 5 
k  1  ( 6k  1)



sinh 
2  3
 sin ( 1)1/ 6  sin ( 1)5/ 6  

cosh
12 5
 2 




sinh 
2  3


cosh
 2  cosh   cos  3 
24 5



1 

 1  12 

k 
k 2 
( 1) k
.99980158731305804717...  
k  0 ( 7 k )!

24611 8
.99984521547922560046... 
 1
165150720 2
.99984999024682965634... 
 (8) 
 ( 1) k
( 1) k 



8 
(4 k  1) 8 
k  1  ( 4 k  1)

J327
1
 (7)  1
 3 
     ( 7 )    
 4
 4 
330301440 
(1) k

8
k  0 ( 2 k  1)


 
=
.99992821782510442180... 
1681 6
 1
933120 3
 ( 1) k
( 1) k 




6
(6k  1) 6 
k  1  ( 6k  1)

J327

( 1) k
6385 9
  ( 9)  
9 
41287680
k  0 ( 2 k  1)

( 1) k
.99997519841274620747...  
k  0 (8k )!
301 7
 h7 
.9999883774094057879... 
524880 3

 (1) k
33367 7
(1) k 

.99998844843872824784...  2 
 1   

7
100776960
(6k  1)7 
k 1  (6k  1)
.99994968418722008982... 
AS 23.2.21
J314
( 1) k
.9999972442680777576...  
k  0 ( 9 k )!

 (1) k
(1) k

.99999727223893094715... 

 1  
8
(6k  1) 8
1410877440 3
k 1  (6k  1)
25743 8




J329
( 1) k
.99999972442680776055...  
k  0 (10k )!
