Mellin Transform and asymptotics: Harmonic sums
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Transcript Mellin Transform and asymptotics: Harmonic sums
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
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Robert Hjalmar Mellin
Born: 1854 in Liminka, Northern Ostrobothnia, Finland
Died: 1933
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Some terminology
• Analytic function
f (s) c0 c1 (s s0 ) c2 (s s0 )2
• Meromorphic function
• Open strip a, b
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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( xv ) x
fundamental strip
f * (s) exists in the strip u, v
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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
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fundamental strip :
u 0 v 1
u, v 0,1
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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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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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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 *
f (s)
s
1
*
,
0
,
,
s
*
f
k k (s)
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0
Basic properties (2/2)
f ( x)
f ( x) log x
,
,
,
f ( x)
sf ( s)
d
f ( x)
dx
( s 1) f * ( s 1) 1, 1
x
0
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f * ( s)
d *
f ( s)
ds
f (t )dt
*
1 *
f ( s 1)
s
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d
x
dx
Zeta function
M k k f ( k x); s
s
*
f
k k (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,
*
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x
M e ; s ( s )( s)
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Some Mellin tranforms
e x
e x 1
e
x2
0,
1, 0
12 s
0,
( s )( s )
1,
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
( s )
( s )
sin s
s sin s
1
s
(1) k k !
( s ) k 1
0,1
1, 0
0,
, k
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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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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
2
s ( s 1)
1
2
s ( s 1)
14
1
2 3( s 1)
( s0 1)
s 1
1 1
1
( s0 0)
2
s
s
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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
1 1
1
(s) 2
1
2 3( s 1)
2
s ( s 1) s 0 s
s
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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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
1
1 1
1
^
2
2
s ( s 1) s 1 s 1 s
s s 0 2 s 1
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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
k
(1) 1
( s ) ^
k! s k
k 0
s
poles of gamma function
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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
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c k
(1) k k !
( s ) k 1
( s )
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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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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.
( s a ) k 1
1
a
Term x at 0
Pole with s.e.
sa
1
a
Term x log x at 0
Pole with s.e.
(s a)2
k
a
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k
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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
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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.)
n0 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
*
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(1) n
n s n
(s )
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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) ^
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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
*
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r
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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)!
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Converse mapping (3/3)
f * (s)
f ( x)
Pole at
left of f.s.
right of f.s
Simple pole
1
left:
s
1
right:
s
Multiple pole
Term in asymptotic expansion x
expansion at 0
expansion at +
left:
1
( s ) k 1
right:
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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!
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Example 4
( s ) s )
s)
, analytic in strip 0,
(1) j ( j 1) 1
s) ^
j!
( )
s j
j 0
f ( x)
1
/ 2i
2 i / 2i
s ,
( s ) x s ds - original function
(1) j ( j 1) j
f ( x)
x O( x M 1 2 ) x 0
j!
( )
j 0
M
f ( x) (1 x) has the same expansion at 0
f ( x) (1 x) ( x), where (x) O( x M ) M 0
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Example 5
( s) (1 s)
sin s
s 0,1
1
( s) ^ (1) n !
sn
n 0
s ,1
n
f ( x)
n
n
(
1)
n
!
x
- the expansion is only asymptotic!
n 0
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M
f ( x) (1) n ! x O( x
n
n
M 1 2
n 0
et
In fact f ( x)
dt
1 xt
0
) x 0
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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)
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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 )
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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
sk
s
Hn
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(1) k Bk 1
log n
k
nk
k 1
log n
1
1
1
4
2n 12n 120n
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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
2
n
(2
n
1)
x
n 1
x x
log( x !) log x e
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B2 n
1
2 n 1
n 1 2n(2n 1) x
2 x
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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!
n0
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)
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x
(1)
n0
1
(
n)
n
2
n!
(1)
n0
n k 1
xn
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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 2 m
*
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
x0
12
240
*
1
1 s
1 1
0 ( s) s ) 0
O( x M ) x 0
2 2
2 x 2
*
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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) s 1 4s k 0 2k 1! s 2k 1
D( x)
36
1
1 2 (2k 1) 2 k 1
( log x
x
x
4 k 0 2k 1!
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The end
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Joint Advanced Student School 2004