Mellin transforms and
asymptotics: Harmonic sums
Phillipe Flajolet,
Xavier Gourdon,
Philippe Dumas
Die Theorie der reziproken
Funktionen und Integrale ist
ein centrales Gebiet, welches
manche anderen Gebiete der
Analysis miteinander verbindet.
Reporter :
Ilya Posov
Saint-Petersburg State University
Faculty of Mathematics and Mechanics
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Joint Advanced Student School 2004
Hjalmar Mellin
Contents
•
•
•
•
2
Mellin transform and its basic properties
Direct mapping
Converse mapping
Harmonic sums
Joint Advanced Student School 2004
Robert Hjalmar Mellin
Born: 1854 in Liminka, Northern Ostrobothnia, Finland
Died: 1933
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Joint Advanced Student School 2004
Some terminology
• Analytic function
f ( s)  c0  c1 ( s  s0 )  c2 ( s  s0 ) 2 
• Meromorphic function
• Open strip  a , b
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Joint Advanced Student School 2004
Mellin transform

M [ f ( x); s ]  f ( s )   f ( x) x dx
s 1
*
0
f ( x)  O( x ) x  0
u
f ( x)  O( x v ) x  
fundamental strip
f * (s) exists in the strip u, v
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Joint Advanced Student School 2004
Mellin transform example
1
f ( x) 
1 x
1
 O(1) x  0
1 x
1
 O( x 1 ) x  
1 x

s 1
x

*
f (s)  
dx 
1 x
sin  s
0
6
fundamental strip :
u  0 v  1
u, v   0,1
Joint Advanced Student School 2004
Gamma function
f ( x)  e
x
e  x  O(1) x  0
e
x
b
 O( x ) b  0 x  

f (s) 
*
e
x
s 1
x dx  ( s )
s( s )  ( s  1)
0
fundamental strip :
u  0 v  
u, v   0, 
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Joint Advanced Student School 2004
Transform of step function
0, x   0,1
H( x)  
1, x  1,  
H( x)  O(1) x  0
b
H( x)  O( x ) b  0 x  

H ( x) 
*

0
1
1
H ( x) x dx   x dx  , s  0, 
s
0
s 1
s 1
1
H( x)  1  H( x), H ( x)   , s  , 0
s
*
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Joint Advanced Student School 2004
Basic properties (1/2)
f ( x)
f (s)
 ,  
x f ( x)
f * ( s  )
  ,   

f (x )
f (1/ x)
f (  x)
 k k f (k x)
9
*
s
f  
 
 f * ( s )
1
*
1

s
*
f (s)
  ,  
 0
  ,  
 ,  
s
*


f
  k k  (s)
Joint Advanced Student School 2004
 0
Basic properties (2/2)
f ( x)
f * (s)
 ,  
f ( x) log x
d *
f (s)
ds
 ,  
f ( x)
 sf ( s )
d
f ( x)
dx
( s  1) f ( s  1)    1,    1

x
0
10
f (t )dt
 ,  
*
*
1 *
 f ( s  1)
s
Joint Advanced Student School 2004
d
x
dx
Zeta function


M   k k f ( k x); s    k k   s f * ( s)
x
e
x
2 x
3 x
g ( x) 

e

e

e
 ...
x
1 e
x
k  1, k  k , f ( x)  e
1 1 1
g (s)   s  s  s 
1 2 3
s  1, 
*
11

x

 M e ; s    ( s )( s)

Joint Advanced Student School 2004
Some Mellin tranforms
e x
( s )
0, 
e x  1
( s )
1, 0
  12 s 
 0, 
 ( s ) ( s )
1, 
e
 x2
1
2
e x
1  e x
1
1 x
log(1  x)
H( x)  10 x 1
x (log x) k H( x)
12

sin  s

s sin  s
1
s
(1) k k !
( s   ) k 1
 0,1
1, 0
0, 
 ,  k 
Joint Advanced Student School 2004
Inversion
Theorem 1
Let f ( x) be integrable with fundamential strip  ,  .
Let   c   and f * (c  i t ) is integrable, then the equality
1
c i 
2 i c i 
f * ( s ) x  s ds  f ( x)
holds almost everywhere.
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Joint Advanced Student School 2004
Laurent expansion
 ( s) 

k
c
(
s

s
)
 k 0
k  r
c r  0
Pole of order r if r>0
Analytic in s0 if r  0
Example :
1
1

 2  3( s  1) 
( s0  1)
2
s ( s  1) s  1
1
1 1
 2  1
( s0  0)
2
s ( s  1) s
s
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Joint Advanced Student School 2004
Singular element
Definition 1
A singular element of  ( s) at s0 is an initial sum of
Laurent expansion truncated at terms of O(1) or smaller :
1
 (s)  2
s ( s  1)

s 0
1 1
 1
2
s
s
1
 s  1  2  3(s  1) 
s 1
 1 1  1 1 
s.e. at s0 =0 :  2   ,  2   1 ,
s s
s 
s
 1   1
  1

s.e. at s0 =  1 : 
, 
 2 , 
 2  3( s  1)  ,

 s  1  s  1   s  1

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Joint Advanced Student School 2004
Singular expansion
Definition 2
Let  ( s) be meromorfic in  with  including all
the poles of  ( s) in . A singular expansion of  ( s )
in  is a formal sum of singular elements of  ( s ) at
all points of 
Notation :  ( s) ^ E ( s  )
1
^
2
s ( s  1)
16
 1 
 1 1
1
 2    
 s  1 
s  s 0  2  s 1
s
s 1
Joint Advanced Student School 2004
s  2, 2
Singular expansion of gamma
function
s( s )  ( s  1)
( s  m  1)
( s ) 
s ( s  1)( s  2) ( s  m)
Thus ( s ) has poles at the points s  m with m 
(1) m 1
( s )
m! s  m

(1) k 1
( s ) ^ 
k! s  k
k 0
s
poles of gamma function
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Joint Advanced Student School 2004
Direct mapping (1/3)
Theorem 2
Let f ( x) have a transform f * ( s) with nonempty fundamental
strip    .
f ( x) 


(  k )A
c k x (log x) k  O( x ) x  0
k  ,      
Then f * ( s) is continuable to a meromorphic function in the
strip     where it admits the singular expansion
f * (s) ^


(  k )A
18
c k
(1) k k !
( s   ) k 1
( s      )
Joint Advanced Student School 2004
Direct mapping (2/3)
f ( x)
f * ( s)
Order at 0: O(x  )
Leftmost boundary of f.s. at ( s )  
Order at +: O(x   )
Rightmost boundary of f.s. at ( s)  
Expansion till O( x ) at 0
Meromorphic continuation till ( s)  
Expansion till O( x ) at + Meromorphic continuation till ( s)  
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Joint Advanced Student School 2004
Direct mapping (3/3)
f ( x)
f * ( s)
Term x a ( log x) k at 0
(1) k k !
Pole with s.e.
( s  a ) k 1
(1) k !
Term x ( log x) at + Pole with s.e. 
k 1
(s  a)
1
a
Term x at 0
Pole with s.e.
sa
1
a
Term x log x at 0
Pole with s.e. 
(s  a)2
k
a
20
k
Joint Advanced Student School 2004
Example 0
f ( x)  e  x
x 2 x3
 1 x   
2! 3!
(1) j j
 
x  O  x M 1 
j!
x 0 j  0
M
f * ( s )  ( s ), s   0, 
f * ( s ) is meromorphically continuable to  M  1, 
j
(

1)
1
*
f (s) ^ 
j! s  j
j 0
M
s   M  1, 
finally
(1) j 1
( s ) ^ 
j! s  j
j 0

s
poles of gamma function
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Joint Advanced Student School 2004
Example 1
x
e
*
f ( x) 
,
f
( s )  ( s ) ( s ) s  1, 
x
1 e

e x
xj
1
  B j 1
, B0  1, B1   ,
x
1  e x  0 j 1
( j  1)!
2

B j 1
1
( s ) ( s) ^ 
j 1 ( j  1)! s  j
(1) j 1
( s ) ^ 
j! s  j
j 0

s
s
1
1
Bm 1
 ( s) ^
s  ,     ,   m)  
s 1
2
m 1
result:   s ) is meromorphic in with the only pole at s0  1
22
Joint Advanced Student School 2004
Example 2

1
f ( x) 
  (1) n x n
1  x x  0 n  0

1/ x
  (1) n 1 x  n
1  1/ x x  n 1
f * ( s) 

, s   0,1
sin  x
(1) n
f ( s) ^ 
, s  ,1 (continuation to the left of the f.s.)
n0 s  n

*
n 1
(

1)
f * ( s ) ^ 
, s  0,  (continuation to the right of the f.s.)
n 1 s  n


f ( s) 
^
sin  x
*
23
(1) n

n s  n
(s  )
Joint Advanced Student School 2004
Example 3 (nonexplicit
transform)
1
1
7
139 6
5473 8
 1  x2  x4 
x 
x  O( x10 ) x  0
4
96
5760
645120
cosh x
f * ( s ) has a fundamental strip 0, 
f ( x) 
1 1 1
7 1
139 1
5473 1




s  10, 
s 4 s  2 96 s  4 5760 s  6 645120 s  8
1 1 1
7 1
139 1
5473 1
*
f (s) ^ 




s
s 4 s  2 96 s  4 5760 s  6 645120 s  8
f * (s) ^
24
Joint Advanced Student School 2004
Converse mapping (1/3)
Theorem 3
Let f ( x)  C(0, ) have a transform f * ( s) with nonempty fundamental
strip    .
Let f * ( s) be meromorphically continuable to     with a finite
number of poles there, and be analytic on ( s)   .
  with r  1 when s   in    .
Let f * ( s)  O s
25
r
Joint Advanced Student School 2004
Converse mapping (2/3)
Theorem 3 (continue)
*
If f ( x) ^


(  k ) A
d  k
1
(s   )k
s     
k  ,      
Then an asymptotic expansion of f ( x) at 0 is
 (  1) k 1 
k 1 
f ( x )   d  k 
x (log x)   O( x )
(  k ) A
 (k  1)!

26
Joint Advanced Student School 2004
Converse mapping (3/3)
f * (s)
f ( x)
Pole at 
Term in asymptotic expansion  x 
left of f.s.
expansion at 0
right of f.s
expansion at +
Simple pole
1
s 
1
right:
s 
Multiple pole
left:
left:
1
( s   ) k 1
right:
27
1
( s   ) k 1
x  at 0
 x  at +
Logarifmic factor
(1) k 
x (log x) k at 0
k!
( 1) k 

x (log x) k at +
k!
Joint Advanced Student School 2004
Example 4
( s )  s )
  s) 
, analytic in strip 0, 
 
(1) j (  j  1) 1
  s) ^ 
j!
( )
s j
j 0

f ( x) 
1
 / 2i 
2 i  / 2i 
s  , 
 ( s ) x  s ds - original function
(1) j (  j  1) j
f ( x)  
x  O( x M 1 2 )
j!
( )
j 0
M
x0
f ( x)  (1  x)  has the same expansion at 0
f ( x)  (1  x)    ( x), where  (x)  O( x M ) M  0
28
Joint Advanced Student School 2004
Example 5

 ( s)  (1  s)
sin  s
s  0,1

1
 ( s) ^  (1) n !
sn
n 0
s  ,1
n

f ( x)
n
n
(

1)
n
!
x
- the expansion is only asymptotic!

n 0






29
M
f ( x)   (1) n ! x  O( x
n
n
M 1 2
n 0

et
In fact f ( x)  
dt
1  xt
0

) x  0




Joint Advanced Student School 2004
Harmonic sums
Definition 3
G ( x)    j g (  k x) harmonic sum
j
j
amplitudes
j
frequencies
 s )    j  j  s
The Dirichlet series
j
M  g (  x ); s     s g * ( s )
(separation property)
G* (s)  (s) g * (s)
30
Joint Advanced Student School 2004
Harmonic sum formula
The Mellin transform of the harmonic sum
G ( x)   k g ( k x) is defined in the intersection
k
of the the fundamental strip of g( x) and the
domain of absolute convergence of the Dirichlet
series ( s)   k k  s (which is of the form
k
( s )   a for some real  a )
G * ( s )   s ) g * ( s )
31
Joint Advanced Student School 2004
Example 6

1   1 x / k
1
1
x
1
h( x )    

, k  ,  k  , g ( x) 

k  x  k 1 k 1  x / k
k
k
1 x
k 1  k
1 1
1
h( n)  H n  1    
2 3
n

 ( s )   k  k
k 1
s

  k 1 s   (1  s )
k 1

h ( s )   ( s ) g ( s )     s )
sin  s
*
*
s  1,0
1
1
 (s) 
   ,  (1  s )    
s 1
s
 1   
*
k  1  k )
h ( s ) ^  2     (1)
s  1, 
s  k 1
sk
s
Hn
32
(1) k Bk 1
log n    
k
nk
k 1
log n   
1
1
1



4
2n 12n 120n
Joint Advanced Student School 2004
Example 7
x

l ( x)  log  x  1)   x     log 1 

n 1  n
1
n  1, n  , g ( x)  x  log(1  x)
n


 s )   k  k
n 1
s

  n s    s )
n 1
l ( s )   s ) g ( s )    s )
*
x 
  s  2,  
n 
*

Simple poles in positive integers
, s  2, 1 Double poles in 0 and -1
s sin  s
 1
1    1
log 2
*
l (s) ^ 



  2s 2
2
(
s

1)

s

1)
s

 
  ( 1) n 1  n)

 n 1 n( s  n)

1
B2 n
1
l ( x)  log( x !)   x  x log x  (1    x  log x  log 2  
2 n 1
2
n 1 2 n(2 n  1) x

x x
log( x !)  log x e
33


B2 n
1
2 n 1
n 1 2n(2n  1) x
2 x  
Joint Advanced Student School 2004
Example 8
e  nx

Lw ( x)   w (polylogarithm Liw ( z )   n 1 z n n  w ) w  k 
n 1 n
1
n  w , k  n, g ( x)  e  x
n
 s )   ( s  k ), Lk * ( s )   ( s  k ) s ) s  1, 


Lk * ( s ) ^
 (1)
1
n   k  n)
n!
n0
n  k 1
(1) k 1 
1
H k 1 




2
s  n (k  1)!  ( s  k  1)
s  k  1
s  , 
(1) k 1 k 1
Lk ( x) 
x [ log x  H k 1 ] 
(k  1)!
L1 2 ( x) 
34

x

  (1)
n0
1

(
 n)
n
2
n!

 (1)
n0
n  k 1
xn
Joint Advanced Student School 2004
n
  k  n)
n!
xn
Example 9
modified theta function

 w ( x)  
n 1
e
 n2 x2
nw
1
 x2
, n  w , n  n, g ( x)  e
n
1 s
 ( s )       s  1) s   0, 
2 2
double pole at s  0 and simple poles at s  2m
*
1
1 1
1   1 1
 (s) ^  2   


s
 s  12 s  2 240 s  4
s
 1 2 1 4
1 ( x)   log x   x 
x 
x0
 12
240
*
1
1 s
1  1
0 ( s)       s) 0 
  O( x M ) x  0
2 2
2 x 2
*
35
Joint Advanced Student School 2004
Example 10

D ( x)   d (k )e  kx , k  d (k ),  k  k , g ( x)  e  x
k 1

 s )   k  k
k 1
s

d (k )
   s    ( s)
k 1 k
D* ( s )   s )  ( s )
2



1

1

(2k  1)
1
 
*
D (s) ^ 

 

2
s  1   4 s  k 0  2k  1! s  2k  1
 ( s  1)
D( x)
36
1
1   2 (2k  1) 2 k 1
(  log x      
x
x
4 k 0  2k  1 !
Joint Advanced Student School 2004
The end
37
Joint Advanced Student School 2004