Theoretical Computer Science Cheat Sheet
Definitions
iff ∃ positive c, n0 such that
0 ≤ f (n) ≤ cg(n) ∀n ≥ n0 .
f (n) = O(g(n))
f (n) = Ω(g(n))
iff ∃ positive c, n0 such that
f (n) ≥ cg(n) ≥ 0 ∀n ≥ n0 .
f (n) = Θ(g(n))
iff f (n) = O(g(n)) and
f (n) = Ω(g(n)).
f (n) = o(g(n))
iff limn→∞ f (n)/g(n) = 0.
iff ∀ > 0, ∃n0 such that
|an − a| < , ∀n ≥ n0 .
lim an = a
n→∞
least b ∈ R such that b ≥ s,
∀s ∈ S.
sup S
greatest b ∈ R such that b ≤
s, ∀s ∈ S.
inf S
lim inf{ai | i ≥ n, i ∈ N}.
lim inf an
Series
n
X
i=
i=1
n(n + 1)
,
2
n
X
i2 =
i=1
n
X
n(n + 1)(2n + 1)
,
6
i3 =
i=1
n2 (n + 1)2
.
4
In general:
n
n
X
X
1
im =
(i + 1)m+1 − im+1 − (m + 1)im
(n + 1)m+1 − 1 −
m+1
i=1
i=1
n−1
m
X
1 X m+1
im =
Bk nm+1−k .
m
+
1
k
i=1
k=0
Geometric series:
n
X
cn+1 − 1
,
ci =
c−1
i=0
n
X
i=0
ici =
c 6= 1,
∞
X
ci =
i=0
ncn+2 − (n + 1)cn+1 + c
,
(c − 1)2
Harmonic series:
n
X
1
,
Hn =
i
i=1
n
X
1
,
1−c
c 6= 1,
∞
X
ci =
i=1
∞
X
ici =
i=0
c
,
1−c
c
,
(1 − c)2
|c| < 1,
|c| < 1.
n(n + 1)
n(n − 1)
Hn −
.
2
4
lim sup an
lim sup{ai | i ≥ n, i ∈ N}.
i=1
n→∞
n→∞
n
n X
X
n+1
i
1
n
Combinations:
Size
k
subHi = (n + 1)Hn − n,
Hn+1 −
.
Hi =
k
m+1
m+1
m
sets of a size n set.
i=1
i=1
n
n X
Stirling numbers (1st kind):
n
n
n
n
n!
k
n
,
2.
1.
=
3.
=
,
=2 ,
Arrangements of an n elek
k
n−k
(n − k)!k!
k
k=0
ment set into k cycles.
n
n−1
n−1
n
n n−1
n
,
5.
=
+
,
4.
=
Stirling
numbers
(2nd
kind):
k k−1
k
k
k−1
k
k
n Partitions of an n element
X
r+k
r+n+1
n
m
n
n−k
=
,
6.
=
,
7.
set into k non-empty sets.
k
n
m
k
k
m−k
n
k=0
n n 1st order Eulerian numbers:
X
X
k
k
n+1
r
s
r+s
=
,
9.
=
,
8.
Permutations π1 π2 . . . πn on
m
m+1
k
n−k
n
k=0
k=0
{1, 2, . . . , n} with k ascents.
k−n−1
n
n
n
n ,
11.
=
= 1,
10.
= (−1)k
2nd order Eulerian numbers.
k
k
1
n
k
Cn
Catalan Numbers: Binary
n
n−1
n−1
n
n−1
−
1,
13.
=
k
+
,
12.
=
2
trees with n + 1 vertices.
k
k
k−1
2
n
n
n
n
n
14.
= (n − 1)!,
15.
= (n − 1)!Hn−1 ,
16.
= 1,
17.
≥
,
1
2
n
k
k
n X
2n
n
1
n
n−1
n−1
n
n
n
,
= n!,
21. Cn =
18.
= (n − 1)
+
,
19.
=
=
,
20.
n+1 n
k
k
k
k−1
n−1
n−1
2
k=0
n
n
n
n
n
n−1
n−1
22.
=
= 1,
23.
=
,
24.
= (k + 1)
+ (n − k)
,
0
n−1
k
n−1−k
k
k
k−1
n
n
n+1
0
n
1 if k = 0,
27.
= 3n − (n + 1)2n +
,
25.
=
26.
= 2n − n − 1,
2
2
k
1
0 otherwise
X
X
n m n X
n
n
x+k
n
n+1
n
k
30. m!
=
,
29.
=
(m + 1 − k)n (−1)k ,
,
28. xn =
m
k
n
m
k
k
n−m
k=0
k=0
k=0
X
n n
n
n
n−k
n
n−k−m
k!,
32.
= 1,
33.
= 0 for n 6= 0,
(−1)
31.
=
0
n
k
m
m
k=0
n X
n
n
n−1
n−1
(2n)n
,
=
34.
= (k + 1)
+ (2n − 1 − k)
,
35.
2n
k
k
k
k−1
k=0
X X
X
n n n
x+n−1−k
n+1
n
k
k
x
,
37.
=
=
(m + 1)n−k ,
36.
=
k
2n
k
m
m
m+1
x−n
n→∞
n→∞
k=0
iHi =
k
k=0
Theoretical Computer Science Cheat Sheet
38.
40.
42.
44.
46.
48.
Identities Cont.
n
X
1 k
,
nn−k = n!
k! m
m
X n k n X
k
Trees
n X
n
x+k
Every tree with n
vertices has n − 1
edges.
Kraft
inequality: If the depths
of the leaves of
a binary tree are
d1 , . . . , dn :
n
X
2−di ≤ 1,
n+1
x
=
39.
=
=
,
m+1
x−n
k m
k
2n
k
k=0
k=0
k=0
X
X n
n
n+1
k+1
k
n
n−k
,
41.
=
(−1)
(−1)m−k ,
=
m
k
m+1
k+1 m
m
k
k
X
X
m
m
n+k
m+n+1
n+k
m+n+1
k
,
43.
=
k(n + k)
,
=
k
m
k
m
k=0
k=0
X
X
n
n
n+1
n+1
k
k
=
=
(−1)m−k , 45. (n − m)!
(−1)m−k , for n ≥ m,
m
m
k+1
m
k+1
m
k k X X m − nm + n m + k n
m−n m+n m+k
n
=
,
47.
=
,
n−m
m+k
n+k
k
m+k
n+k
k
n−m
k
X k X k
k n−k n
n−k
n
n
`+m
n
`+m
,
49.
=
.
=
`
m
k
`+m
`
m
k
`
`+m
`
k
i=1
and equality holds
only if every internal node has 2
sons.
k
Recurrences
Master method:
T (n) = aT (n/b) + f (n),
a ≥ 1, b > 1
logb a−
If ∃ > 0 such that f (n) = O(n
then
T (n) = Θ(nlogb a ).
)
If f (n) = Θ(nlogb a ) then
T (n) = Θ(nlogb a log2 n).
If ∃ > 0 such that f (n) = Ω(nlogb a+ ),
and ∃c < 1 such that af (n/b) ≤ cf (n)
for large n, then
T (n) = Θ(f (n)).
Substitution (example): Consider the
following recurrence
i
Ti+1 = 22 · Ti2 , T1 = 2.
Note that Ti is always a power of two.
Let ti = log2 Ti . Then we have
ti+1 = 2i + 2ti , t1 = 1.
Let ui = ti /2i . Dividing both sides of
the previous equation by 2i+1 we get
2i
ti
ti+1
=
+ i.
i+1
i+1
2
2
2
Substituting we find
u1 = 12 ,
ui+1 = 12 + ui ,
which is simply ui = i/2. So we find
i−1
that Ti has the closed form Ti = 2i2 .
Summing factors (example): Consider
the following recurrence
T (n) = 3T (n/2) + n, T (1) = 1.
Rewrite so that all terms involving T
are on the left side
T (n) − 3T (n/2) = n.
Now expand the recurrence, and choose
a factor which makes the left side “telescope”
1 T (n) − 3T (n/2) = n
3 T (n/2) − 3T (n/4) = n/2
..
..
..
.
.
.
log2 n−1
3
T (2) − 3T (1) = 2
Let m = log2 n. Summing the left side
we get T (n) − 3m T (1) = T (n) − 3m =
T (n) − nk where k = log2 3 ≈ 1.58496.
Summing the right side we get
m−1
m−1
X n
X i
i
3
3
=
n
.
2
i
2
i=0
i=0
Let c = 32 . Then we have
m
m−1
X
c −1
i
c =n
n
c−1
i=0
Multiply
X
X
X and sum:
gi+1 xi =
2gi xi +
xi .
i≥0
= 2n(clog2 n − 1)
= 2n(c(k−1) logc n − 1)
= 2nk − 2n,
and so T (n) = 3n − 2n. Full history recurrences can often be changed to limited
history ones (example): Consider
i−1
X
Ti = 1 +
Tj , T0 = 1.
j=0
Ti+1 = 1 +
i
X
Tj .
j=0
Subtracting we find
i
i−1
X
X
Tj − 1 −
Tj
Ti+1 − Ti = 1 +
j=0
= Ti .
And so Ti+1 = 2Ti = 2i+1 .
i≥0
i≥0
P
We choose G(x) = i≥0 xi gi . Rewrite
in terms of G(x):
X
G(x) − g0
= 2G(x) +
xi .
x
i≥0
k
Note that
Generating functions:
1. Multiply both sides of the equation by xi .
2. Sum both sides over all i for
which the equation is valid.
3. Choose a generatingPfunction
∞
G(x). Usually G(x) = i=0 xi gi .
3. Rewrite the equation in terms of
the generating function G(x).
4. Solve for G(x).
5. The coefficient of xi in G(x) is gi .
Example:
gi+1 = 2gi + 1, g0 = 0.
j=0
Simplify:
1
G(x)
= 2G(x) +
.
x
1−x
Solve for G(x):
x
.
G(x) =
(1 − x)(1 − 2x)
Expand this using partial fractions:
1
2
−
G(x) = x
1 − 2x 1 − x


X
X
2i xi −
xi 
= x 2
=
X
i≥0
i+1
(2
i≥0
So gi = 2i − 1.
i≥0
− 1)x
i+1
.
Theoretical Computer Science Cheat Sheet
π ≈ 3.14159,
e ≈ 2.71828,
γ ≈ 0.57721,
φ=
√
1+ 5
2
i
2i
pi
General
1
2
2
4
2
3
Bernoulli Numbers (Bi = 0, odd i 6= 1):
3
4
8
16
5
7
5
6
32
64
11
13
7
8
9
128
256
512
17
19
23
10
11
1,024
2,048
29
31
12
13
4,096
8,192
37
41
14
15
16,384
32,768
43
47
16
17
18
65,536
131,072
262,144
53
59
61
19
20
524,288
1,048,576
67
71
21
22
2,097,152
4,194,304
73
79
23
24
8,388,608
16,777,216
83
89
25
26
27
33,554,432
67,108,864
134,217,728
97
101
103
28
29
30
268,435,456
536,870,912
1,073,741,824
107
109
113
31
32
2,147,483,648
4,294,967,296
127
131
Pascal’s Triangle
1
11
121
1331
14641
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
− 21 ,
≈ 1.61803,
1
6,
1
− 30
,
1
1
+ 16 + 24
+ 120
+ ···
n
x
= ex .
lim 1 +
n→∞
n
n
n+1
.
1 + n1 < e < 1 + n1
1
11e
e
n
+
−O
1 + n1 = e −
.
2
2n 24n
n3
Harmonic numbers:
25 137 49 363 761 7129
1, 32 , 11
6 , 12 , 60 , 20 , 140 , 280 , 2520 , . . .
1
2
ln n < Hn < ln n + 1,
1
Hn = ln n + γ + O
.
n
Factorial, Stirling’s approximation:
1, 2, 6, 24, 120, 720, 5040, 40320, 362880,
...
n n
1
2πn
.
1+Θ
e
n
Ackermann’s
 function and inverse:
i=1
 2j
a(i, j) = a(i − 1, 2)
j=1

a(i − 1, a(i, j − 1)) i, j ≥ 2
n! =
√
1− 5
2
≈ −.61803
Probability
B2 =
B4 =
B0 = 1, B1 =
1
1
5
.
B6 = 42 , B8 = − 30 , B10 = 66
Change of base, quadratic formula:
√
−b ± b2 − 4ac
loga x
,
.
logb x =
loga b
2a
Euler’s number e:
e=1+
φ̂ =
√
α(i) = min{j | a(j, j) ≥ i}.
Binomial distribution:
n k n−k
,
q = 1 − p,
Pr[X = k] =
p q
k
n
X
n k n−k
k
p q
= np.
[X]
=
E
k
k=1
Poisson distribution:
e−λ λk
, E[X] = λ.
Pr[X = k] =
k!
Normal (Gaussian) distribution:
2
2
1
e−(x−µ) /2σ , E[X] = µ.
p(x) = √
2πσ
The “coupon collector”: We are given a
random coupon each day, and there are n
different types of coupons. The distribution of coupons is uniform. The expected
number of days to pass before we to collect all n types is
nHn .
Continuous distributions: If
Z b
p(x) dx,
Pr[a < X < b] =
a
then p is the probability density function of
X. If
Pr[X < a] = P (a),
then P is the distribution function of X. If
P and p both exist then
Z a
p(x) dx.
P (a) =
−∞
Expectation: If X is discrete
X
g(x) Pr[X = x].
E[g(X)] =
x
If X continuous
Z
Z ∞ then
g(x)p(x)
dx
=
[g(X)]
=
E
−∞
∞
g(x) dP (x).
−∞
Variance, standard deviation:
VAR[X] = E[X 2 ] − E[X]2 ,
p
σ = VAR[X].
For events A and B:
Pr[A ∨ B] = Pr[A] + Pr[B] − Pr[A ∧ B]
Pr[A ∧ B] = Pr[A] · Pr[B],
iff A and B are independent.
Pr[A ∧ B]
Pr[A|B] =
Pr[B]
For random variables X and Y :
E[X · Y ] = E[X] · E[Y ],
if X and Y are independent.
E[X + Y ] = E[X] + E[Y ],
E[cX] = c E[X].
Bayes’ theorem:
Pr[B|Ai ] Pr[Ai ]
.
Pr[Ai |B] = Pn
j=1 Pr[Aj ] Pr[B|Aj ]
Inclusion-exclusion:
n
n
i X
h_
Xi =
Pr[Xi ] +
Pr
i=1
i=1
n
X
X
(−1)k+1
k=2
ii <···<ik
Pr
k
h^
i
Xij .
j=1
Moment inequalities:
1
Pr |X| ≥ λ E[X] ≤ ,
λ
i
h
1
Pr X − E[X] ≥ λ · σ ≤ 2 .
λ
Geometric distribution:
q = 1 − p,
Pr[X = k] = pq k−1 ,
∞
X
1
kpq k−1 = .
E[X] =
p
k=1
Theoretical Computer Science Cheat Sheet
Trigonometry
Matrices
More Trig.
C
Multiplication:
(0,1)
b
C
A
θ
(-1,0)
C = A · B,
(1,0)
c
a
B
Pythagorean theorem:
C 2 = A2 + B 2 .
(0,-1)
cos a = B/C,
sec a = C/B,
cos a
B
cot a =
= .
sin a
A
circle:
AB
.
A+B+C
cos x =
1 + cot2 x = csc2 x,
cos x = − cos(π − x),
sin x = sin(π − x),
tan x = cot π2 − x ,
cot x = − cot(π − x),
csc x = cot x2 − cot x,
sin(x ± y) = sin x cos y ± cos x sin y,
cos(x ± y) = cos x cos y ∓ sin x sin y,
cos 2x = cos2 x − sin2 x,
cos 2x = 1 − 2 sin2 x,
n
XY
π i=1
Hyperbolic Functions
Definitions:
ex − e−x
,
sinh x =
2
x
−x
e −e
tanh x = x
,
e + e−x
1
,
sech x =
cosh x
−x
x
e +e
,
2
1
,
csch x =
sinh x
1
coth x =
.
tanh x
cosh x =
Identities:
cosh2 x − sinh2 x = 1,
tanh2 x + sech2 x = 1,
coth2 x − csch2 x = 1,
sinh(−x) = − sinh x,
tanh(−x) = − tanh x,
cosh(x + y) = cosh x cosh y + sinh x sinh y,
2 tan x
,
1 + tan2 x
cos 2x = 2 cos2 x − 1,
sin 2x =
cos 2x =
1 − tan2 x
,
1 + tan2 x
cos(x + y) cos(x − y) = cos2 x − sin2 y.
iπ
e
= −1.
c
v2.02 1994
by Steve Seiden
[email protected]
http://www.csc.lsu.edu/~seiden
sinh 2x = 2 sinh x cosh x,
cosh 2x = cosh2 x + sinh2 x,
cosh x + sinh x = ex ,
cosh x − sinh x = e−x ,
(cosh x + sinh x)n = cosh nx + sinh nx,
2 sinh2 x2
= cosh x − 1,
θ
sin θ
0
0
π
6
π
4
π
3
π
2
1
2
√
2
2
√
3
2
1
cos θ
1
tan θ
0
√
√
0
1
√
3
∞
3
2
√
2
2
1
2
3
3
2 cosh2 x2
a
A
c
Law of cosines:
B
c2 = a2 +b2 −2ab cos C.
Area:
A = 12 hc,
= 12 ab sin C,
c2 sin A sin B
.
2 sin C
Heron’s formula:
=
sa = s − a,
sb = s − b,
sc = s − c.
ai,π(i) .
sinh(x + y) = sinh x cosh y + cosh x sinh y,
cot2 x − 1
2 tan x
,
cot 2x =
tan 2x =
2 ,
2 cot x
1 − tan x
sin(x + y) sin(x − y) = sin2 x − sin2 y,
Euler’s equation:
eix = cos x + i sin x,
Permanents:
h
√
A = s · sa · sb · sc ,
s = 12 (a + b + c),
aei + bf g + cdh
− ceg − f ha − ibd.
=
cosh(−x) = cosh x,
tan x ± tan y
,
1 ∓ tan x tan y
cot x cot y ∓ 1
,
cot(x ± y) =
cot x ± cot y
tan(x ± y) =
sin 2x = 2 sin x cos x,
2 × 2 and 3 × 3 determinant:
a b
c d = ad − bc,
a b c d e f = gb c − ha c + ia b
e f d f d e
g h i 1
,
sec x
sin2 x + cos2 x = 1,
1 + tan2 x = sec2 x,
sin x = cos π2 − x ,
b
Determinants: det A 6= 0 iff A is non-singular.
det A · B = det A · det B,
n
XY
sign(π)ai,π(i) .
det A =
perm A =
Identities:
1
,
sin x =
csc x
1
,
tan x =
cot x
ai,k bk,j .
k=1
π i=1
Definitions:
sin a = A/C,
csc a = C/A,
A
sin a
= ,
tan a =
cos a
B
Area, radius of inscribed
1
2 AB,
ci,j =
(cos θ, sin θ)
n
X
n ∈ Z,
= cosh x + 1.
. . . in mathematics
you don’t understand things, you
just get used to
them.
– J. von Neumann
More identities:
r
1 − cos x
x
,
sin 2 =
2
r
1 + cos x
,
cos x2 =
2
r
1 − cos x
,
tan x2 =
1 + cos x
1 − cos x
,
=
sin x
sin x
,
=
1 + cos x
r
1 + cos x
,
cot x2 =
1 − cos x
1 + cos x
,
=
sin x
sin x
,
=
1 − cos x
eix − e−ix
,
sin x =
2i
eix + e−ix
,
cos x =
2
eix − e−ix
,
tan x = −i ix
e + e−ix
e2ix − 1
,
= −i 2ix
e +1
sinh ix
,
sin x =
i
cos x = cosh ix,
tanh ix
.
tan x =
i
Theoretical Computer Science Cheat Sheet
Number Theory
The Chinese remainder theorem: There exists a number C such that:
C ≡ r1
.. ..
. .
mod m1
..
.
C ≡ rn mod mn
if mi and mj are relatively prime for i 6= j.
Euler’s function: φ(x) is the number of
positive integers
Qnless than x relatively
prime to x. If i=1 pei i is the prime factorization of x then
n
Y
piei −1 (pi − 1).
φ(x) =
i=1
Euler’s theorem: If a and b are relatively
prime then
1 ≡ aφ(b) mod b.
Fermat’s theorem:
1 ≡ ap−1 mod p.
The Euclidean algorithm: if a > b are integers then
gcd(a, b) = gcd(a mod b, b).
Qn
If i=1 pei i is the prime factorization of x
then
n
Y
X
piei +1 − 1
.
d=
S(x) =
pi − 1
i=1
d|x
Perfect Numbers: x is an even perfect number iff x = 2n−1 (2n −1) and 2n −1 is prime.
Wilson’s theorem: n is a prime iff
(n − 1)! ≡ −1 mod n.
Möbius 
inversion:
1
if i = 1.


0
if i is not square-free.
µ(i) =
r
if
i is the product of
(−1)


r distinct primes.
If
G(a) =
X
F (d),
d|a
then
F (a) =
X
d|a
a
.
µ(d)G
d
Prime numbers:
ln ln n
pn = n ln n + n ln ln n − n + n
ln n
n
,
+O
ln n
n
2!n
n
+
+
π(n) =
ln n (ln n)2
(ln n)3
n
+O
.
(ln n)4
Graph Theory
Definitions:
Loop
An edge connecting a vertex to itself.
Directed
Each edge has a direction.
Simple
Graph with no loops or
multi-edges.
Walk
A sequence v0 e1 v1 . . . e` v` .
Trail
A walk with distinct edges.
Path
A trail with distinct
vertices.
Connected
A graph where there exists
a path between any two
vertices.
Component A maximal connected
subgraph.
Tree
A connected acyclic graph.
Free tree
A tree with no root.
DAG
Directed acyclic graph.
Eulerian
Graph with a trail visiting
each edge exactly once.
Hamiltonian Graph with a cycle visiting
each vertex exactly once.
Cut
A set of edges whose removal increases the number of components.
Cut-set
A minimal cut.
Cut edge
A size 1 cut.
k-Connected A graph connected with
the removal of any k − 1
vertices.
k-Tough
∀S ⊆ V, S 6= ∅ we have
k · c(G − S) ≤ |S|.
k-Regular
A graph where all vertices
have degree k.
k-Factor
A k-regular spanning
subgraph.
Matching
A set of edges, no two of
which are adjacent.
Clique
A set of vertices, all of
which are adjacent.
Ind. set
A set of vertices, none of
which are adjacent.
Vertex cover A set of vertices which
cover all edges.
Planar graph A graph which can be embeded in the plane.
Plane graph An embedding of a planar
graph.
X
deg(v) = 2m.
v∈V
If G is planar then n − m + f = 2, so
f ≤ 2n − 4, m ≤ 3n − 6.
Any planar graph has a vertex with degree ≤ 5.
Notation:
E(G) Edge set
V (G) Vertex set
c(G)
Number of components
G[S]
Induced subgraph
deg(v) Degree of v
∆(G) Maximum degree
δ(G)
Minimum degree
χ(G) Chromatic number
χE (G) Edge chromatic number
Complement graph
Gc
Complete graph
Kn
Kn1 ,n2 Complete bipartite graph
r(k, `) Ramsey number
Geometry
Projective coordinates: triples
(x, y, z), not all x, y and z zero.
(x, y, z) = (cx, cy, cz) ∀c 6= 0.
Cartesian
Projective
(x, y)
(x, y, 1)
y = mx + b (m, −1, b)
x=c
(1, 0, −c)
Distance formula, Lp and L∞
metric:
p
(x1 − x0 )2 + (y1 − y0 )2 ,
1/p
|x1 − x0 |p + |y1 − y0 |p
,
1/p
.
lim |x1 − x0 |p + |y1 − y0 |p
p→∞
Area of triangle (x0 , y0 ), (x1 , y1 )
and (x2 , y2 ):
x1 − x0 y1 − y0 1
.
2 abs x − x
2
0 y2 − y0
Angle formed by three points:
(x2 , y2 )
`2
θ
(x1 , y1 )
`1
(0, 0)
(x1 , y1 ) · (x2 , y2 )
cos θ =
.
`1 `2
Line through two points (x0 , y0 )
and (x1 , y1 ):
x y 1
x0 y0 1 = 0.
x1 y1 1 Area of circle, volume of sphere:
A = πr2 ,
V = 43 πr3 .
If I have seen farther than others,
it is because I have stood on the
shoulders of giants.
– Issac Newton
Theoretical Computer Science Cheat Sheet
π
Calculus
Wallis’ identity:
2 · 2 · 4 · 4 · 6 · 6···
π =2·
1 · 3 · 3 · 5 · 5 · 7···
Brouncker’s continued fraction expansion:
12
π
4 =1+
32
2+
52
2+
Gregrory’s series:
π
1
4 =1− 3 +
1
5
−
1
7
2+
72
2+···
+
1
9
− ···
Derivatives:
1.
du
d(cu)
=c ,
dx
dx
4.
du
d(un )
= nun−1 ,
dx
dx
7.
du
d(cu )
= (ln c)cu ,
dx
dx
9.
du
d(sin u)
= cos u ,
dx
dx
Newton’s series:
1
1
1·3
π
6 = 2 + 2 · 3 · 23 + 2 · 4 · 5 · 25 + · · ·
Sharp’s series:
π
6
1
1
1
1 + 2
− 3
+···
= √ 1− 1
3 ·3 3 ·5 3 ·7
3
Euler’s series:
π2
6
π2
8
π2
12
=
=
=
1
12
1
12
1
12
+
+
−
1
22
1
32
1
22
+
+
+
1
32
1
52
1
32
1
42
1
72
1
42
+
+
−
+
+
+
1
52
1
92
1
52
+ ···
+ ···
− ···
Partial Fractions
Let N (x) and D(x) be polynomial functions of x.
We can break down
N (x)/D(x) using partial fraction expansion. First, if the degree of N is greater
than or equal to the degree of D, divide
N by D, obtaining
N 0 (x)
N (x)
= Q(x) +
,
D(x)
D(x)
where the degree of N 0 is less than that of
D. Second, factor D(x). Use the following rules: For a non-repeated factor:
A
N 0 (x)
N (x)
=
+
,
(x − a)D(x)
x−a
D(x)
where
N (x)
A=
D(x)
1 dk N (x)
.
Ak =
k! dxk D(x) x=a
The reasonable man adapts himself to the
world; the unreasonable persists in trying
to adapt the world to himself. Therefore
all progress depends on the unreasonable.
– George Bernard Shaw
14.
1
du
d(arcsec u)
= √
,
2
dx
u 1 − u dx
du
d(sinh u)
= cosh u ,
21.
dx
dx
23.
du
d(tanh u)
= sech2 u ,
dx
dx
25.
du
d(sech u)
= − sech u tanh u ,
dx
dx
20.
26.
1
du
d(arcsinh u)
=√
,
2
dx
1 + u dx
1 du
d(arctanh u)
=
,
29.
dx
1 − u2 dx
32.
Z
n 6= −1,
4.
sin x dx = − cos x,
d(arccsc u)
−1
du
= √
,
2
dx
u 1 − u dx
d(cosh u)
du
22.
= sinh u ,
dx
dx
d(coth u)
du
= − csch2 u ,
dx
dx
d(csch u)
du
= − csch u coth u ,
dx
dx
d(arccsch u)
−1
du
√
=
.
dx
|u| 1 + u2 dx
Z
(u + v) dx =
v dx,
Z
tan x dx = − ln | cos x|,
11.
Z
cot x dx = ln | cos x|,
Z
sec x dx = ln | sec x + tan x|,
Z
14.
Z
u dx +
Z
1
dx = ln x,
5.
ex dx = ex ,
x
Z
Z
du
dv
7.
u dx = uv − v dx,
dx
dx
Z
9.
cos x dx = sin x,
Z
12.
d(cot u)
du
= csc2 u ,
dx
dx
Z
2.
dx
= arctan x,
1 + x2
10.
d(cos u)
du
= − sin u ,
dx
dx
d(arccosh u)
1
du
=√
,
2
dx
u − 1 dx
d(arccoth u)
1 du
30.
= 2
,
dx
u − 1 dx
−1
du
d(arcsech u)
= √
,
dx
u 1 − u2 dx
Integrals:
Z
Z
1.
cu dx = c u dx,
Z
d(ln u)
1 du
=
,
dx
u dx
28.
31.
8.
du
d(ecu )
= cecu ,
dx
dx
d(csc u)
du
= − cot u csc u ,
dx
dx
24.
27.
6.
6.
,
16.
19.
Z
d(uv)
dv
du
=u
+v ,
dx
dx
dx
d(arccos u)
−1 du
=√
,
dx
1 − u2 dx
d(arccot u)
−1 du
18.
=
,
dx
1 + u2 dx
1
du
d(arcsin u)
=√
,
2
dx
1 − u dx
1 du
d(arctan u)
=
,
17.
dx
1 + u2 dx
3.
12.
15.
x=a
k=0
where
du
d(sec u)
= tan u sec u ,
dx
dx
1
xn+1 ,
x dx =
n+1
dv
dx
10.
13.
n
3.
8.
du
d(tan u)
= sec2 u ,
dx
dx
Z
For a repeated factor:
m−1
X
Ak
N 0 (x)
N (x)
=
,
+
m
m−k
(x − a) D(x)
(x − a)
D(x)
d(u + v)
du dv
=
+
,
dx
dx dx
v du
d(u/v)
dx − u
5.
=
dx
v2
11.
.
2.
arcsin xa dx = arcsin xa +
p
a2 − x2 ,
13.
a > 0,
csc x dx = ln | csc x + cot x|,
Theoretical Computer Science Cheat Sheet
Z
15.
arccos
Z
17.
Z
19.
Z
x
a dx
= arccos
sin2 (ax)dx =
1
2a
x
a
−
p
Calculus Cont.
a2
−
x2 ,
Z
16.
a > 0,
arctan xa dx = x arctan xa −
Z
ax − sin(ax) cos(ax) ,
18.
cos2 (ax)dx =
1
2a
a
2
a > 0,
ax + sin(ax) cos(ax) ,
Z
sec2 x dx = tan x,
ln(a2 + x2 ),
20.
csc2 x dx = − cot x,
Z
Z
Z
sinn−1 x cos x n − 1
cosn−1 x sin x n − 1
+
sinn−2 x dx,
+
cosn−2 x dx,
22.
cosn x dx =
n
n
n
n
Z
Z
Z
tann−1 x
cotn−1 x
n
n−2
n
− tan
− cotn−2 x dx, n 6= 1,
x dx, n 6= 1,
24.
cot x dx = −
tan x dx =
n−1
n−1
Z
tan x secn−1 x n − 2
+
secn−2 x dx, n 6= 1,
secn x dx =
n−1
n−1
Z
Z
Z
cot x cscn−1 x n − 2
+
cscn−2 x dx, n 6= 1,
27.
sinh x dx = cosh x,
28.
cosh x dx = sinh x,
cscn x dx = −
n−1
n−1
Z
Z
Z
tanh x dx = ln | cosh x|, 30.
coth x dx = ln | sinh x|, 31.
sech x dx = arctan sinh x, 32.
csch x dx = ln tanh x2 ,
sinn x dx = −
21.
Z
23.
Z
25.
Z
26.
Z
29.
Z
33.
2
sinh x dx =
1
4
sinh(2x) −
Z
36.
arcsinh
x
a dx
= x arcsinh
x
a
Z
1
2 x,
−
34.
2
cosh x dx =
1
4
sinh(2x) +
1
2 x,
Z
35.
sech2 x dx = tanh x,
Z
p
x2
+
a2 ,
a > 0,
37.
arctanh xa dx = x arctanh xa +
a
2
ln |a2 − x2 |,

x p
 x arccosh − x2 + a2 , if arccosh xa > 0 and a > 0,
a
38.
arccosh xa dx =
p
 x arccosh x + x2 + a2 , if arccosh x < 0 and a > 0,
a
a
Z
p
dx
√
= ln x + a2 + x2 , a > 0,
39.
a2 + x2
Z p
Z
p
2
dx
1
x
=
arctan
,
a
>
0,
41.
a2 − x2 dx = x2 a2 − x2 + a2 arcsin xa , a > 0,
40.
a
a
2
2
a +x
Z
p
4
42.
(a2 − x2 )3/2 dx = x8 (5a2 − 2x2 ) a2 − x2 + 3a8 arcsin xa , a > 0,
Z
Z
Z
a + x
1
x
dx
dx
dx
x
,
√
ln
= arcsin a , a > 0,
44.
=
= √
,
43.
45.
2
2
2
2
3/2
2
2
2
a −x
2a
a−x
(a − x )
a −x
a a2 − x2
Z
Z p
p
p
p
2
dx
√
47.
a2 ± x2 dx = x2 a2 ± x2 ± a2 ln x + a2 ± x2 ,
= ln x + x2 − a2 , a > 0,
46.
x2 − a2
Z
Z
√
1 x 2(3bx − 2a)(a + bx)3/2
dx
,
49.
x a + bx dx =
=
ln
,
48.
2
ax + bx
a
a + bx
15b2
√
√ Z
Z
Z √
√
a + bx − a x
1
a + bx
1
√
√
dx = 2 a + bx + a
dx,
51.
dx = √ ln √
50.
√ , a > 0,
x
x a + bx
a + bx
a + bx + a 2
Z p
Z √ 2
a + √a2 − x2 p
a − x2
2
2
53.
x a2 − x2 dx = − 13 (a2 − x2 )3/2 ,
52.
dx = a − x − a ln ,
x
x
Z
Z
a + √a2 − x2 p
p
4
dx
√
55.
= − a1 ln 54.
x2 a2 − x2 dx = x8 (2x2 − a2 ) a2 − x2 + a8 arcsin xa , a > 0,
,
2
2
x
a −x
Z
Z
2
p
p
2
x dx
x dx
x
√
√
56.
= − a2 − x2 ,
57.
= − x2 a2 − x2 + a2 arcsin a,
a > 0,
2
2
2
2
a −x
a
−
x
√
√
√
Z
Z
a + a2 + x2 p
p
a2 + x2
x2 − a2
a
59.
dx = a2 + x2 − a ln dx = x2 − a2 − a arccos |x|
, a > 0,
58.
,
x
x
x
Z
Z p
dx
x
,
√
√
61.
= a1 ln 60.
x x2 ± a2 dx = 13 (x2 ± a2 )3/2 ,
x x2 + a2
a + a2 + x2 Z
Theoretical Computer Science Cheat Sheet
Z
62.
Z
64.
Z
66.
Z
67.
Z
68.
Calculus Cont.
√
x2 ± a2
dx
dx
1
a
√
√
,
= a arccos |x| , a > 0,
63.
=∓
a2 x
x x2 − a2
x2 x2 ± a2
√
Z
p
x2 ± a2
(x2 + a2 )3/2
x dx
√
= x2 ± a2 ,
65.
dx
=
∓
,
x4 3a2 x3
x2 ± a2

√
2ax + b − b2 − 4ac 
1


√
√
ln , if b2 > 4ac,

2
2
dx
b − 4ac
2ax + b + b − 4ac =
ax2 + bx + c 

2ax + b
2

√
arctan √
,
if b2 < 4ac,
4ac − b2
4ac − b2

√ p
1


√ ln 2ax + b + 2 a ax2 + bx + c , if a > 0,

a
dx
√
=
−2ax − b
1
ax2 + bx + c 

√
,
if a < 0,
arcsin √
−a
b2 − 4ac
Z
p
dx
2ax + b p 2
4ax − b2
√
,
ax2 + bx + c dx =
ax + bx + c +
2
4a
8a
ax + bx + c
Z
√
Z
b
ax2 + bx + c
dx
√
−
,
69.
2
a
2a
ax + bx + c
√ √

−1 2 c ax2 + bx + c + bx + 2c 


,
Z
 √c ln x
dx
√
=
70.
2

x ax + bx + c 
1
bx + 2c

 √ arcsin √
,
−c
|x| b2 − 4ac
Z
p
2 2
a )(x2 + a2 )3/2 ,
71.
x3 x2 + a2 dx = ( 13 x2 − 15
Z
Z
72.
Z
73.
x dx
√
=
2
ax + bx + c
xn sin(ax) dx = − a1 xn cos(ax) +
n
x cos(ax) dx =
Z
74.
xn eax dx =
Z
75.
xn eax
−
a
sin(ax) −
Z
n
a
xn ln(ax) dx = xn+1
Z
76.
1 n
ax
xn (ln ax)m dx =
n
a
if c > 0,
if c < 0,
Z
E f (x) = f (x + 1).
Fundamental Theorem:
X
f (x) = ∆F (x) ⇔
f (x)δx = F (x) + C.
b
X
n−1
x
1
ln(ax)
−
n+1
(n + 1)2
n+1
x
m
(ln ax)m −
n+1
n+1
f (x)δx =
b−1
X
a
Differences:
∆(cu) = c∆u,
f (i).
i=a
∆(u + v) = ∆u + ∆v,
∆(uv) = u∆v + E v∆u,
∆(xn ) = nxn−1 ,
∆(Hx ) = x−1 ,
∆(2x ) = 2x ,
x
x
∆ m
= m−1
.
∆(cx ) = (c − 1)cx ,
Sums:
P
P
cu δx = c u δx,
P
P
P
(u + v) δx =
u δx + v δx,
P
P
u∆v δx = uv − E v∆u δx,
n+1
P −1
P n
x δx = Hx ,
x δx = xm+1 ,
P
P x
x
x
x
c
,
c δx = c−1
m δx = m+1 .
Falling Factorial Powers:
xn = x(x − 1) · · · (x − n + 1), n > 0,
xn =
Z
sin(ax) dx,
1
,
(x + 1) · · · (x + |n|)
n < 0,
xn+m = xm (x − m)n .
Rising Factorial Powers:
xn−1 eax dx,
xn = x(x + 1) · · · (x + n − 1),
n > 0,
x0 = 1,
,
Z
xn =
xn (ln ax)m−1 dx.
x1 =
x2 =
x1
x2 + x1
=
=
x1
x2 − x1
x3 =
x4 =
x3 + 3x2 + x1
4
x + 6x3 + 7x2 + x1
=
=
x3 − 3x2 + x1
4
x − 6x3 + 7x2 − x1
x5 =
x5 + 15x4 + 25x3 + 10x2 + x1
=
x5 − 15x4 + 25x3 − 10x2 + x1
x1 =
x2 =
x1
x + x1
x1 =
x2 =
x1
x − x1
x3 =
x4 =
x3 + 3x2 + 2x1
x4 + 6x3 + 11x2 + 6x1
x3 =
x4 =
x3 − 3x2 + 2x1
x4 − 6x3 + 11x2 − 6x1
x5 =
x5 + 10x4 + 35x3 + 50x2 + 24x1
x5 =
x5 − 10x4 + 35x3 − 50x2 + 24x1
2
Difference, shift operators:
∆f (x) = f (x + 1) − f (x),
x0 = 1,
xn−1 cos(ax) dx,
n
a
Finite Calculus
2
1
,
(x − 1) · · · (x − |n|)
n < 0,
xn+m = xm (x + m)n .
Conversion:
xn = (−1)n (−x)n = (x − n + 1)n
= 1/(x + 1)−n ,
xn = (−1)n (−x)n = (x + n − 1)n
= 1/(x − 1)−n ,
n n X
n k X n
x =
(−1)n−k xk ,
xn =
k
k
k=1
k=1
n X
n
(−1)n−k xk ,
xn =
k
k=1
n X
n k
n
x .
x =
k
k=1
Theoretical Computer Science Cheat Sheet
Series
Taylor’s series:
∞
X (x − a)i
(x − a)2 00
f (a) + · · · =
f (i) (a).
f (x) = f (a) + (x − a)f (a) +
2
i!
i=0
Expansions:
∞
X
1
= 1 + x + x2 + x3 + x4 + · · ·
=
xi ,
1−x
i=0
∞
X
1
= 1 + cx + c2 x2 + c3 x3 + · · ·
=
ci xi ,
1 − cx
i=0
∞
X
1
n
2n
3n
=
1
+
x
+
x
+
x
+
·
·
·
=
xni ,
1 − xn
i=0
∞
X
x
2
3
4
= x + 2x + 3x + 4x + · · ·
=
ixi ,
(1 − x)2
i=0
∞
X
1
dn
in xi ,
= x + 2n x2 + 3n x3 + 4n x4 + · · · =
xk n
dx
1−x
i=0
∞
X
xi
,
= 1 + x + 12 x2 + 16 x3 + · · ·
=
ex
i!
i=0
∞
X
xi
=
(−1)i+1 ,
ln(1 + x)
= x − 12 x2 + 13 x3 − 14 x4 − · · ·
i
i=1
∞
i
Xx
1
= x + 12 x2 + 13 x3 + 14 x4 + · · ·
,
=
ln
1−x
i
i=1
∞
X
x2i+1
1 3
1 5
1 7
,
(−1)i
sin x
= x − 3! x + 5! x − 7! x + · · · =
(2i + 1)!
i=0
∞
X
x2i
1 2
1 4
1 6
,
x + 4!
x − 6!
x + ··· =
(−1)i
cos x
= 1 − 2!
(2i)!
i=0
∞
X
x2i+1
,
= x − 13 x3 + 15 x5 − 17 x7 + · · ·
=
(−1)i
tan−1 x
(2i + 1)
i=0
∞ X
n i
2
= 1 + nx + n(n−1)
x
+
·
·
·
=
(1 + x)n
x,
2
i
i=0
∞ X
i+n i
1
n+2 2
= 1 + (n + 1)x + 2 x + · · · =
x,
(1 − x)n+1
i
i=0
∞
X
Bi xi
x
1
1 2
1
4
=
1
−
,
x
+
x
−
x
+
·
·
·
=
2
12
720
x
e −1
i!
i=0
∞
X
√
2i i
1
1
2
3
(1 − 1 − 4x)
= 1 + x + 2x + 5x + · · ·
=
x,
2x
i+1 i
i=0
∞ X
2i i
1
√
= 1 + x + 2x2 + 6x3 + · · ·
=
x,
i
1 − 4x
i=0
√
n
∞ X
1 − 1 − 4x
2i + n i
1
4+n 2
√
= 1 + (2 + n)x + 2 x + · · · =
x,
2x
i
1 − 4x
i=0
∞
X
1
1
25 4
3
ln
= x + 32 x2 + 11
x
+
x
+
·
·
·
=
Hi xi ,
6
12
1−x 1−x
i=1
2
∞
X
Hi−1 xi
1
1
4
,
= 12 x2 + 34 x3 + 11
x
+
·
·
·
=
ln
24
2
1−x
i
i=2
∞
X
x
2
3
4
= x + x + 2x + 3x + · · ·
=
Fi xi ,
1 − x − x2
i=0
∞
X
Fn x
2
3
=
F
x
+
F
x
+
F
x
+
·
·
·
=
Fni xi .
n
2n
3n
1 − (Fn−1 + Fn+1 )x − (−1)n x2
i=0
0
Ordinary power series:
∞
X
ai xi .
A(x) =
i=0
Exponential power series:
∞
X
xi
ai .
A(x) =
i!
i=0
Dirichlet power series:
∞
X
ai
.
A(x) =
ix
i=1
Binomial theorem:
n X
n n−k k
y .
x
(x + y)n =
k
k=0
Difference of like powers:
n−1
X
xn−1−k y k .
xn − y n = (x − y)
k=0
For ordinary power series:
∞
X
(αai + βbi )xi ,
αA(x) + βB(x) =
i=0
xk A(x) =
A(x) −
Pk−1
i=0
xk
∞
X
ai−k xi ,
i=k
i
ai x
A(cx) =
=
∞
X
∞
X
ai+k xi ,
i=0
ci ai xi ,
i=0
∞
X
(i + 1)ai+1 xi ,
A0 (x) =
i=0
xA0 (x) =
∞
X
iai xi ,
i=1
Z
A(x) dx =
∞
X
ai−1
i=1
A(x) + A(−x)
=
2
A(x) − A(−x)
=
2
∞
X
i
xi ,
a2i x2i ,
i=0
∞
X
a2i+1 x2i+1 .
i=0
Pi
Summation: If bi = j=0 ai then
1
A(x).
B(x) =
1−x
Convolution:


∞
i
X
X

aj bi−j  xi .
A(x)B(x) =
i=0
j=0
God made the natural numbers;
all the rest is the work of man.
– Leopold Kronecker
Theoretical Computer Science Cheat Sheet
Series
Expansions:
1
1
ln
n+1
(1 − x)
1−x
xn
1
ln
1−x
n
tan x
1
ζ(x)
ζ(x)
ζ 2 (x)
Escher’s Knot
−n
∞ X
n+i i
1
i
=
(Hn+i − Hn )
x,
=
xi ,
i
x
n
i=0
i=0
∞ ∞ X
X
n i
i n!xi
x
n
,
=
(e − 1)
=
x,
i!
i
n
i=0
i=0
∞ ∞
X
X
i n!xi
(−4)i B2i x2i
,
x cot x
=
,
=
(2i)!
n i!
i=0
i=0
∞
∞
2i 2i
2i−1
X
X
1
i−1 2 (2 − 1)B2i x
,
ζ(x)
=
(−1)
,
=
x
(2i)!
i
i=1
i=1
∞
∞
X
X
µ(i)
φ(i)
ζ(x − 1)
=
=
,
,
x
i
ζ(x)
ix
i=1
i=1
Y
1
=
,
Stieltjes Integration
1 − p−x
p
If G is continuous in the interval [a, b] and F is nondecreasing then
∞
X
Z b
P
d(i)
=
where d(n) = d|n 1,
G(x) dF (x)
xi
∞
X
i=1
ζ(x)ζ(x − 1)
=
∞
X
S(i)
xi
where S(n) =
P
d|n
d,
i=1
2n−1
ζ(2n)
x
sin x
√
n
1 − 1 − 4x
2x
x
e sin x
|B2n | 2n
π , n ∈ N,
=
(2n)!
∞
X
(4i − 2)B2i x2i
=
,
(−1)i−1
(2i)!
i=0
2
=
=
√
1−x
x
2
arcsin x
x
∞
X
n(2i + n − 1)!
i=0
∞
X
i=1
s
1−
exists. If a ≤ b ≤ c then
Z
Z c
G(x) dF (x) =
=
∞
X
i=0
=
∞
X
i=0
i!(n + i)!
i/2
2
sin
i!
iπ
4
xi ,
i
x,
(4i)!
√
xi ,
i
16 2(2i)!(2i + 1)!
4i i!2
x2i .
(i + 1)(2i + 1)!
Cramer’s Rule
If we have equations:
a1,1 x1 + a1,2 x2 + · · · + a1,n xn = b1
a2,1 x1 + a2,2 x2 + · · · + a2,n xn = b2
..
..
..
.
.
.
an,1 x1 + an,2 x2 + · · · + an,n xn = bn
Let A = (ai,j ) and B be the column matrix (bi ). Then
there is a unique solution iff det A 6= 0. Let Ai be A
with column i replaced by B. Then
det Ai
.
xi =
det A
Improvement makes strait roads, but the crooked
roads without Improvement, are roads of Genius.
– William Blake (The Marriage of Heaven and Hell)
a
a
Z
b
a
Z
b
G(x) d F (x) + H(x) =
a
Z
b
Z
c · G(x) dF (x) =
a
G(x) dF (x).
b
If the integrals involved exist
Z
Z b
G(x) + H(x) dF (x) =
a
c
G(x) dF (x) +
Z
b
a
Z
b
G(x) d c · F (x) = c
a
Z
b
b
G(x) dF (x) +
a
b
H(x) dF (x),
a
Z
b
G(x) dF (x) +
G(x) dH(x),
a
Z b
G(x) dF (x),
Z
G(x) dF (x) = G(b)F (b) − G(a)F (a) −
a
a
b
F (x) dG(x).
a
If the integrals involved exist, and F possesses a derivative F 0 at every
point in [a, b] then
Z b
Z b
G(x) dF (x) =
G(x)F 0 (x) dx.
a
00 47 18 76 29 93 85 34 61 52
86 11 57 28 70 39 94 45 02 63
95 80 22 67 38 71 49 56 13 04
59 96 81 33 07 48 72 60 24 15
73 69 90 82 44 17 58 01 35 26
68 74 09 91 83 55 27 12 46 30
37 08 75 19 92 84 66 23 50 41
14 25 36 40 51 62 03 77 88 99
a
Fibonacci Numbers
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .
Definitions:
Fi = Fi−1 +Fi−2 , F0 = F1 = 1,
F−i = (−1)i−1 Fi ,
Fi = √15 φi − φ̂i ,
21 32 43 54 65 06 10 89 97 78
Cassini’s identity: for i > 0:
42 53 64 05 16 20 31 98 79 87
Fi+1 Fi−1 − Fi2 = (−1)i .
Additive rule:
The Fibonacci number system:
Every integer n has a unique
representation
n = Fk1 + Fk2 + · · · + Fkm ,
where ki ≥ ki+1 + 2 for all i,
1 ≤ i < m and km ≥ 2.
Fn+k = Fk Fn+1 + Fk−1 Fn ,
F2n = Fn Fn+1 + Fn−1 Fn .
Calculation by matrices:
n
0 1
Fn−2 Fn−1
.
=
Fn−1
Fn
1 1