Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and...
Transcript of Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and...
Mellin transforms and asymptotics: Harmonic sums
Phillipe Flajolet,Xavier Gourdon,Philippe Dumas
Die Theorie der reziprokenFunktionen und Integrale ist
ein centrales Gebiet, welchesmanche anderen Gebiete der
Analysis miteinander verbindet.
Hjalmar MellinReporter :Ilya PosovSaint-Petersburg State UniversityFaculty of Mathematics and Mechanics
Joint Advanced Student School 20041
Contents
• Mellin transform and its basic properties• Direct mapping• Converse mapping• Harmonic sums
Joint Advanced Student School 20042
Robert Hjalmar Mellin
Born: 1854 in Liminka, Northern Ostrobothnia, FinlandDied: 1933
Joint Advanced Student School 20043
Some terminology
• Analytic function
• Meromorphic function
• Open strip
20 1 0 2 0( ) ( ) ( )f s c c s s c s s= + − + − +L
,a b⟨ ⟩
Joint Advanced Student School 20044
Mellin transform* 1
0
[ ( ); ] ( ) ( ) sM f x s f s f x x dx∞
−= = ∫u( ) O( ) 0f x x x= →v( ) O( )f x x x= → +∞
*( ) exists in the strip ,f s u v⟨− − ⟩
fundamental strip
Joint Advanced Student School 20045
Mellin transform example
1
1*
0
1( )1
1 O(1) 01
1 O( )1
( )1 sin
s
f xx
xx
x xx
xf s dxx s
ππ
−
+∞ −
=+
= →+
= → +∞+
= =+∫
fundamental strip :0 1, 0,1
u vu v= = −
⟨− − ⟩ = ⟨ ⟩
Joint Advanced Student School 20046
Gamma function
* 1
0
( )O(1) 0O( ) 0
( ) ( ) ( ) ( 1)
x
x
x b
x s
f x ee xe x b x
f s e x dx s s s s
−
−
− −
+∞− −
=
= →
= ∀ > → +∞
= = Γ Γ = Γ +∫
0, 0,
u vu v= = −∞
⟨− − ⟩ = ⟨ +∞⟩
fundamental strip :
Joint Advanced Student School 20047
Transform of step function
[ ]( )
1* 1 1
0 0
*
0, 0,1H( )
1, 1,
H( ) O(1) 0H( ) O( ) 0
1H ( ) ( ) , 0,
1H( ) 1 H( ), H ( ) , ,0
b
s s
xx
x
x xx x b x
x H x x dx x dx ss
x x x ss
−
+∞− −
⎧ ∈⎪= ⎨ ∈ +∞⎪⎩= →
= ∀ > → +∞
= = = ∈⟨ +∞⟩
= − = − ∈⟨−∞ ⟩
∫ ∫
Joint Advanced Student School 20048
Basic properties (1/2)
( )
*
*
*
*
*
*
( ) ( ) ,( ) ( ) ,
1( ) , 0
(1/ ) ( ) ,1( ) ( ) , 0
( ) ( )
s
sk k kk k
f x f sx f x f s
sf x f
f x f s
f x f s
f x f s
ν
ρ
α βν α ν β ν
ρα ρβ ρρ ρ
β α
µ α β µµ
λ µ λ µ−
⟨ ⟩+ ⟨ − − ⟩
⎛ ⎞⟨ ⟩ >⎜ ⎟
⎝ ⎠− − ⟨− − ⟩
⟨ ⟩ >
∑ ∑
Joint Advanced Student School 20049
Basic properties (2/2)*
*
*
*
*
0
( ) ( ) ,
( ) log ( ) ,
( ) ( ) ,
( ) ( 1) ( 1) 1, 1
1( ) ( 1)x
f x f sdf x x f sds
df x sf s xdx
d f x s f sdx
f t dt f ss
α β
α β
α β
α β
⟨ ⟩
⟨ ⟩
′ ′Θ − ⟨ ⟩ Θ =
′ ′− − − ⟨ + + ⟩
− +∫
Joint Advanced Student School 200410
Zeta function
( ) *
2 3
*
( ); ( )
( ) ...1
1, , ( )1 1 1( ) ; ( ) ( )1 2 3
1,
sk k kk k
xx x x
x
xk k
xs s s
M f x s f s
eg x e e ee
k f x e
g s M e s s s
s
λ µ λ µ
λ µ
ζ
−
−− − −
−
−
−
⎡ ⎤ =⎣ ⎦
= = + + +−
= = =
⎛ ⎞ ⎡ ⎤= + + + = Γ⎜ ⎟ ⎣ ⎦⎝ ⎠∈⟨ +∞⟩
∑ ∑
L
Joint Advanced Student School 200411
Some Mellin tranforms
( )2 1 12 2
0 1
1
( ) 0,1 ( ) 1,0
0,
( ) ( ) 1,1
1 0,11 sin
log(1 ) 1,0sin
1H( ) 1 0,
( 1) !(log ) H( ) ,( )
x
x
x
x
x
x
kk
k
e se s
e se s s
e
x s
xs s
xs
kx x x ks
α
ζ
ππππ
αα
−
−
−
−
−
< <
+
Γ ⟨ +∞⟩− Γ ⟨− ⟩
Γ ⟨ +∞⟩
Γ ⟨ +∞⟩−
⟨ ⟩+
+ ⟨− ⟩
≡ ⟨ +∞⟩
−⟨− +∞⟩ ∈
+Joint Advanced Student School 200412
Inversion
Theorem 1
*
i*
i
Let ( ) be integrable with fundamential strip , .Let and ( i ) is integrable, then the equality
1 ( ) ( )2 iholds almost everywhere.
cs
c
f xc f c t
f s x ds f x
α β
α β
π
+ ∞−
− ∞
⟨ ⟩
< < +
=∫
Joint Advanced Student School 200413
Laurent expansion
0
0
02
02 2
( ) ( )
0Pole of order if r>0Analytic in if 0Example :
1 1 2 3( 1) ( 1)( 1) 11 1 1 1 ( 0)
( 1)
kk
k r
r
s c s s
cr
s r
s ss s s
ss s s s
φ+∞
≥−
−
= −
≠
≤
= + + + + = −+ +
= − + + =+
∑
L
L
Joint Advanced Student School 200414
Singular elementDefinition 1
0
2 20 1
0 2 2
A singular element of ( ) at is an initial sum ofLaurent expansion truncated at terms of O(1) or smaller :
1 1 1 1( ) 1 2 3( 1)( 1) 1
1 1 1 1s.e. at =0 : , 1
s s
s s
s ss s s s s
ss s s s
φ
φ→ →−
= − + − + + + ++ +
⎡ ⎤ ⎡ ⎤− − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
= =L L
0
,
1 1 1s.e. at = 1 : , 2 , 2 3( 1) ,1 1 1
s ss s s⎡ ⎤ ⎡ ⎤ ⎡ ⎤− + + + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦
K
K
Joint Advanced Student School 200415
Singular expansionDefinition 2
2
Let ( ) be meromorfic in with including all the poles of ( ) in . A singular expansion of ( )in is a formal sum of singular elements of ( ) atall points of Notation : ( ) ( )
1 1( 1)
ss s
s
s E s
s s
φφ φ
φ
φ
Ω ℘Ω
Ω℘
∈Ω
+
^
^ 21 0 1
1 1 1 2,21 2s s s
ss s s=− = =
⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ − + ∈⟨− ⟩⎢ ⎥ ⎢ ⎥ ⎢ ⎥+⎣ ⎦ ⎣ ⎦ ⎣ ⎦
Joint Advanced Student School 200416
Singular expansion of gamma function
0
( ) ( 1)( 1)( )
( 1)( 2) ( )Thus ( ) has poles at the points with
( 1) 1( )!( 1) 1( )
!
m
k
k
s s ss ms
s s s s ms s m m
sm s m
s sk s k
+∞
=
Γ = Γ +Γ + +
Γ =+ + +
Γ = − ∈
−Γ
+−
Γ ∈+∑
L
^
poles of gamma function
Joint Advanced Student School 200417
Direct mapping (1/3)Theorem 2
*
( )
*
Let ( ) have a transform ( ) with nonempty fundamentalstrip .
( ) (log ) O( ) 0
,Then ( ) is continuable to a meromorphic function in thestrip where it admi
kk
k A
f x f s
f x c x x x x
kf s
ξ γξ
ξ
α β
γ ξ α
γ β
+,
, ∈
⟨ , ⟩
= + →
∈ − < − ≤
⟨− , ⟩
∑
*1
( )
ts the singular expansion( 1) !( ) ( )
( )
k
k kk A
kf s c ssξ
ξ
γ βξ, +
, ∈
−∈⟨− , ⟩
+∑^
Joint Advanced Student School 200418
Direct mapping (2/3)*( ) ( )
Order at 0: O( ) Leftmost boundary of f.s. at ( )Order at + : O( ) Rightmost boundary of f.s. at ( )Expansion till O( ) at 0 Meromorphic continuation till ( )Expansion till O( )
f x f sx s
x sx sx
α
β
γ
δ
αβγ
−
−
ℜ =∞ ℜ =
ℜ ≥ −at + Meromorphic continuation till ( )s δ∞ ℜ ≤ −
Joint Advanced Student School 200419
Direct mapping (3/3)*
1
1
2
( ) ( )( 1) !Term ( log ) at 0 Pole with s.e.
( )( 1) !Term ( log ) at + Pole with s.e.
( )1Term at 0 Pole with s.e.
1Term log at 0 Pole with s.e. ( )
ka k
k
ka k
k
a
a
f x f skx x
s akx x
s a
xs a
x xs a
+
+
−+
−∞ −
+
+
−+
Joint Advanced Student School 200420
Example 0
( )2 3
1
0 0
*
*
M*
0
0
( 1)( ) 1 O2! 3! !
( ) ( ), 0,( ) is meromorphically continuable to 1,
( 1) 1( ) 1,!
finally( 1) 1( )
!
jMx j M
x j
j
j
j
j
x xf x e x x xj
f s s sf s M
f s s Mj s j
s sj s j
− +
→ =
=
+∞
=
−= = − + − + = +
= Γ ∈⟨ +∞⟩
⟨− − +∞⟩
−∈⟨− − +∞⟩
+
−Γ ∈
+
∑
∑
∑
L
^
^
poles of gamma functionJoint Advanced Student School 200421
Example 1*
1 0 10 1
1
1
0
1
( ) , ( ) ( ) ( ) 1,1
1, 1, ,1 ( 1)! 2
1( ) ( )( 1)!
( 1) 1( )!
1 1( ) , , )1 2 1
result: ) is meromorph
x
x
x j
jxx j
j
j
j
j
m
ef x f s s s se
e xB B Be j
Bs s s
j s j
s sj s j
Bs s ms m
s
ζ
ζ
ζ ζ ζ
ζ
−
−
− +∞
+−→ =−
+∞+
=−
+∞
=
+
= = Γ ∈⟨ +∞⟩−
= = = −− +
Γ ∈+ +
−Γ ∈
+
∈ (0) = − (− = −− +(
∑
∑
∑
L
^
^
^
0ic in with the only pole at 1s =
Joint Advanced Student School 200422
0 0
1
1
*
*
0
1*
1
1( ) ( 1)1
1/ ( 1)1 1/
( ) , 0,1sin
( 1)( ) , ,1 (continuation to the left of the f.s.)
( 1)( ) , 0, (continuation to the ri
n n
x n
n n
x n
n
n
n
n
f x xx
x xx
f s sx
f s ss n
f s ss n
ππ
+
+∞
→ =
+∞− −
→+∞ =
+∞
=
−+∞
=
= = −+
= −+
= ∈⟨ ⟩
−∈⟨−∞ ⟩
+
−− ∈⟨ +∞⟩
−
∑
∑
∑
∑
^
^
*
ght of the f.s.)
( 1)( ) ( )sin
n
nf s s
x s nππ ∈
−≡ ∈
+∑^
Example 2
Joint Advanced Student School 200423
Example 3 (nonexplicit transform)
2 4 6 8 10
*
*
*
1 1 7 139 5473( ) 1 O( ) 04 96 5760 645120cosh
( ) has a fundamental strip 0,1 1 1 7 1 139 1 5473 1( ) 10,
4 2 96 4 5760 6 645120 81 1 1 7 1 139 1 5473 1( )
4 2 96 4 5760 6 645120
f x x x x x x xx
f s
f s ss s s s s
f ss s s s
= = − + − + + →
⟨ +∞⟩
− + − + ∈⟨− +∞⟩+ + + +
− + − ++ + +
^
^8
ss
+ ∈+
L
Joint Advanced Student School 200424
Converse mapping (1/3)Theorem 3
*
*
*
Let ( ) C(0, ) have a transform ( ) with nonempty fundamentalstrip .Let ( ) be meromorphically continuable to with a finitenumber of poles there, and be analytic on ( ) .
Let (
f x f s
f ss
f s
α β
γ βγ
∈ +∞⟨ , ⟩
⟨− , ⟩ℜ = −
( )) O with 1 when in .rs r s γ β−= > →+∞ ⟨− , ⟩
L
Joint Advanced Student School 200425
Converse mapping (2/3)Theorem 3 (continue)
*
( )
11
( )
1If ( )( )
,Then an asymptotic expansion of ( ) at 0 is
( 1)( ) (log ) O( )( 1)!
k kk A
kk
kk A
f x d ss
kf x
f x d x x xk
ξξ
ξ γξ
ξ
γ βξ
γ ξ β
,, ∈
−− −
,, ∈
∈⟨− , ⟩−
∈ − < − ≤
⎛ ⎞−= +⎜ ⎟−⎝ ⎠
∑
∑
L
^
Joint Advanced Student School 200426
Converse mapping (3/3)*( ) ( )
Pole at Term in asymptotic expansion left of f.s. expansion at 0 right of f.s expansion at +Simple pole
1 left: at 0
1 right: at +
Multiple pole Logarifmi
f s f xx
xs
xs
ξ
ξ
ξ
ξ
ξ
ξ
−
−
−
≈
∞
−
− ∞−
1
1
c factor1 ( 1) left: (log ) at 0
( ) !1 ( 1) right: (log ) at +
( ) !
kk
k
kk
k
x xs k
x xs k
ξ
ξ
ξ
ξ
−+
−+
−−
−− ∞
−
Joint Advanced Student School 200427
Example 4
0
/ 2 i
/ 2 i
1 2
0
( ) )) , analytic in strip 0,
( 1) ( 1) 1) ,! ( )
1( ) ( ) - original function2 i
( 1) ( 1)( ) O( ) 0! ( )
( ) (1 ) has the
j
j
s
jMj M
j
s ss
js sj s j
f x s x ds
jf x x x xj
f x x
ν
ν
ν
νφ νν
νφ νν
φπ
νν
+∞
=
+ ∞−
− ∞
+
=
−
Γ Γ( −( = ⟨ ⟩
Γ( )
− Γ + −( ∈⟨−∞ ⟩
Γ +
=
− Γ + −= + →
Γ
= +
∑
∫
∑
^
same expansion at 0( ) (1 ) ( ), where ( ) O( ) 0Mf x x x x x Mν ϖ ϖ−= + + = ∀ >
Joint Advanced Student School 200428
Example 5
0
0
1 2
0
0
( ) (1 ) 0,1sin
1( ) ( 1) ! ,1
( ) ( 1) ! - the expansion is only asymptotic!
( ) ( 1) ! O( ) 0
In fact ( )1
n
n
n n
n
Mn n M
n
t
s s ss
s n ss n
f x n x
f x n x x x
ef x dtxt
πφπ
φ+∞
=
+∞
=
+
=
+∞ −
= Γ − ∈⟨ ⟩
− ∈⟨−∞ ⟩+
−
⎛ ⎞= − + →⎜ ⎟⎝ ⎠⎛ ⎞
=⎜ ⎟+⎝ ⎠
∑
∑
∑
∫
^
Joint Advanced Student School 200429
Harmonic sumsDefinition 3
[ ] *
* *
( ) ( ) harmonic sum
amplitudes
frequencies
) The Dirichlet series
( ); ( )
( ) ( ) ( )
j kj
j
j
sj j
j
s
G x g x
s
M g x s g s
G s s g s
λ µ
λ
µ
λ µ
µ µ
−
−
=
Λ( =
=
= Λ
∑
∑(separation property)
Joint Advanced Student School 200430
Harmonic sum formula
k
The Mellin transform of the harmonic sum( ) ( ) is defined in the intersection
of the the fundamental strip of g( ) and thedomain of absolute convergence of the Dirichletseries ( ) (w
k kk
sk k
G x g x
x
s
λ µ
λ µ −
=
Λ =
∑
∑
* *
hich is of the form
( ) for some real )
( ) ) ( )a as
G s s g s
σ σℜ >
= Λ(
Joint Advanced Student School 200431
Example 61 1
1
1 1
* *
*2
1 1 1 / 1 1( ) , , , ( )1 / 1
1 1 1( ) 12 3
( ) (1 )
( ) ( ) ( ) ) 1,0sin
1 1( ) , (1 )1
1( )
k kk k
n
s sk k
k k
x k xh x g xk k x k x k k k x
h n Hn
s k s
h s s g s s ss
s ss s
h ss s
λ µ
λ µ ζ
πζπ
ζ γ ζ γ
γ
+∞ +∞
= =
+∞ +∞− − +
= =
⎡ ⎤= − = = = =⎢ ⎥+ + +⎣ ⎦
= = + + + +
Λ = = = −
= Λ = − (1− ∈⟨− ⟩
= + + − = − +−⎡ ⎤−⎢ ⎥⎣ ⎦
∑ ∑
∑ ∑
L
L
^1
41
1 )( 1) 1,
( 1) 1 1 1 1log log2 12 120
k
k
kk
n kk
k ss k
BH n nk n n n n
ζ
γ γ
+∞
=
≥
( −− − ∈⟨− +∞⟩
−
−+ + + + − + +
∑
∑ L
Joint Advanced Student School 200432
Example 7
1
1 1
* *
*2 2
( ) log 1) log 1 2,
11, , ( ) log(1 )
) )
( ) ) ( ) ) , 2, 1sin
1 1 1 log 2 (( )( 1) 1) 2
n
n n
s sk k
n n
x xl x x x sn n
g x x xn
s n s
l s s g s s ss s
l ss s s s
γ
λ µ
λ µ ζ
πζπ
γ π
+∞
=
+∞ +∞−
= =
⎡ ⎤⎛ ⎞= Γ( + − = − + ∈< − +∞ >⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
= = = − +
Λ( = = = (−
= Λ( = − (− ∈⟨− − ⟩
⎡ ⎤⎡ ⎤−+ + − +⎢ ⎥⎢ ⎥+ ( +⎣ ⎦ ⎣ ⎦
∑
∑ ∑
^
( )
1
1
22 1
1
22 1
1
1) )( )
1 1( ) log( !) log (1 log log 22 2 (2 1)
1log( !) log 22 (2 1)
n
n
nn
n
x x nn
n
nn s n
Bl x x x x x x xn n x
Bx x e xn n x
ζ
γ γ π
π
−+∞
=
+∞
−=
+∞−
−=
− (−+
= − = − − ) − + +−
= +−
∑
∑
∑
Simple poles in positive integersDouble poles in 0 and -1
Joint Advanced Student School 200433
Example 81
1
*
1* 1
20
1
( ) (polylogarithm ( ) )
1 , , ( )
) ( ), ( ) ( ) ) 1,
) 1 ( 1) 1( ) ( 1)! ( 1)! ( 1) 1
nxn w
w ww nn
xn kw
k
kn k
kn
n k
eL x Li z z n w kn
n g x en
s s k L s s k s s
k n HL sn s n k s k s k
λ µ
ζ ζ
ζ
−+∞+∞ −=
=
−
−+∞−
=≠ −
= = = ∈
= = =
Λ( = + = + Γ( ∈⟨ +∞⟩
⎡ ⎤( − −− + +⎢ ⎥+ − + − + −⎣ ⎦
∑ ∑
∑^
11
10
1
12
1 20
,( 1) )( ) [ log ] ( 1)( 1)! !
( )( ) ( 1)
!
kk n n
k kn
n k
n n
n
sk nL x x x H x
k n
nL x x
x n
ζ
ζπ
− +∞−
−=≠ −
+∞
=
∈⟨−∞ +∞⟩
− ( −= − + + −
−
−= + −
∑
∑Joint Advanced Student School 200434
Example 9modified theta function
2 22
1
*1
*1 2
2 41
*0
1( ) , , , ( )
1( ) 1) 0,2 2
double pole at 0 and simple poles at 21 1 1 1 1( )
12 2 240 41 1( ) log 0
12 2401( )
n xx
w n nw wn
ex n g x en nss s s
s s m
s ss s s s
x x x x x
s
λ µ
ζ
γ
γ
−+∞−
=
Θ = = = =
⎛ ⎞Θ = Γ ( + ∈⟨ +∞⟩⎜ ⎟⎝ ⎠
= = −
⎡ ⎤Θ + + + + ∈⎢ ⎥2 + +⎣ ⎦
Θ = − + + + + →2
Θ =
∑
L
L
^
01 1) O( ) 0
2 2 2 2Ms s x x
xπζ⎛ ⎞Γ ( Θ = − + →⎜ ⎟
⎝ ⎠Joint Advanced Student School 200435
Example 10
( )
( )
1
1 1
*
2*
20
22 1
0
( ) ( ) , ( ), , ( )
( )) ( )
( ) ) ( )
1 1 ( 2 1) 1( )( 1) 1 4 2 1 ! 2 1
1 1 ( 2 1)( ) ( log4 2 1 !
kx xk k
k
sk k s
k k
k
k
k
D x d k e d k k g x e
d ks sk
D s s s
kD ss s s k s k
kD x x xx k
λ µ
λ µ ζ
ζ
γ ζ
ζγ
+∞− −
=
+∞ +∞− 2
−= =
2
+∞
=
+∞+
=
= = = =
Λ( = = =
= Γ(
⎡ ⎤ − −⎡ ⎤+ + −⎢ ⎥ ⎢ ⎥− − + + +⎣ ⎦⎣ ⎦− −
− + ) + −+
∑
∑ ∑
∑
∑
^
Joint Advanced Student School 200436
The end
Joint Advanced Student School 200437