Mellin transforms and asymptotics: Harmonic sums

Transcript

1 Mellin tranform and aymptotic: Harmonic um Phillipe Flajolet, Xavier Gourdon, Philippe Duma Die Theorie der reziproen Funtionen und Integrale it ein centrale Gebiet, welche manche anderen Gebiete der Analyi miteinander verbindet. Reporter : Ilya Poov Saint-Peterburg State Univerity Faculty of Mathematic and Mechanic Hjalmar Mellin

2 Content Mellin tranform and it baic propertie Direct mapping Convere mapping Harmonic um 2

3 Robert Hjalmar Mellin Born: 854 in Limina, Northern Otrobothnia, Finland Died: 933 3

4 Some terminology Analytic function f( ) c c ( ) c ( ) 2 = L Meromorphic function Open trip ab, 4

5 Mellin tranform M[ f( x); ] f ( ) f( x) x dx f x x x = = u ( ) = O( ) v f( x) O( x ) x = + f ( ) exit in the trip u, v fundamental trip 5

6 Mellin tranform example f( x) = + x = O() + x x = O( x ) + x x + + x f dx () π = = + x inπ fundamental trip : u = v= u, v =, 6

7 Gamma function f( x) e x = e x = O() x x b e = O( x ) b> x + + x f e x dx () = =Γ() Γ () =Γ ( + ) fundamental trip : u = v= u, v =, + 7

8 Tranform of tep function [ ], x, H( x) =, x, + H( x) = O() x ( ) b H( x) = O( x ) b> x + + H( x) = H( x) x dx= x dx=,, + H( x) = H( x), H ( x) =,, 8

9 Baic propertie (/2) f x f ( ) ( ), ν x f x f ( ) ( ), f x ρ ρ ρα ρβ ρ > f(/ x) f ( ) β, α ρ ( ) f, f x f µ ( µ ) ( ) α, β µ ( ) f( x) f ( ) λ µ λ µ αβ + ν α ν β ν > 9

10 Baic propertie (2/2) f x f ( ) ( ), d f x x f d αβ ( )log ( ), αβ Θ Θ= f ( x) f ( ) α, β x dx d d f ( x ) ( ) f ( ) α +, β + dx x f() t dt f ( + )

11 Zeta function ( ) M λ f( µ x); = λ µ f ( ) e gx e e e e x λ =, µ =, f( x) = e x x 2x 3x ( ) = = x x g () = L M e ; ζ () () = Γ 2 3, +

12 Some Mellin tranform 2 e e x e x x 2 Γ (), + Γ ( ), 2 2 ( ) Γ, + x e ζ () Γ (), + x e π, + x inπ π log( + x), inπ H( x) < x<, + α ( )! x (log x) H( x) α, + + ( + α)

13 Theorem Inverion Let f( x) be integrable with fundamential trip α, β. Let α < c< β and f ( c+ i t) i integrable, then the equality 2π i c+ i c i f x d = f x () () hold almot everywhere. 3

14 Laurent expanion φ() = c ( ) c r + r Pole of order r if r> Analytic in if r Example : = ( + ) + L ( = ) ( + ) + = + + L ( = ) ( + ) 4

15 Definition Singular element A ingular element of φ( ) at i an initial um of Laurent expanion truncated at term of O() or maller : φ() = = + L = ( + ) + L ( + ) e. at = :, + 2 2, K.e. at = :, + 2, ( + ), K 5

16 Definition 2 2 Singular expanion Let φ( ) be meromorfic in Ω with including all the pole of φ( ) in Ω. A ingular expanion of φ( ) in Ω i a formal um of ingular element of φ( ) at all point of Notation : φ( ) ^ E ( Ω) ^ 2, 2 2 ( + ) = = = 6

17 Singular expanion of gamma Γ () =Γ ( + ) = function Γ ( + m+ ) Γ () = ( + )( + 2) L( + m) Thu Γ ( ) ha pole at the point = m with m Γ() Γ() ^ m ( ) m! + m + ( )! + pole of gamma function 7

18 Theorem 2 Direct mapping (/3) Let ( ) have a tranform f x f trip αβ,. ξ γ f( x) = c x (log x) + O( x ) x ( ξ, ) A, γ < ξ α Then f ξ, ( ) with nonempty fundamental ( ) i continuable to a meromorphic function in the trip γ, β where it admit the ingular expanion ( )! () ( γ, β ) f ^ cξ, ( ξ, ) A + ( + ξ ) + 8

19 Direct mapping (2/3) f x f ( ) ( ) α Order at : O( x ) Leftmot boundary of f.. at R ( ) = α β Order at + : O( x ) Rightmot boundary of f.. at R ( ) = β γ Expanion till O( x ) at Meromorphic continuation till R( ) γ δ Expanion till O( x ) at + Meromorphic continuation till R( ) δ 9

20 Direct mapping (3/3) f x f ( ) ( ) a ( )! Term x (log x) at Pole with.e. ( + a ) a Term x (log x) at + Pole with.e. a Term x at Pole with.e. a Term x log x at Pole with.e. + ( )! ( + a) + a ( + a) + 2 2

21 Example x x ( ) f( x) = e = x+ + L = x + O x 2! 3! j! 2 3 M j x j M+ () =Γ(),, + f x j= f M M j f ^ M j= j= ( ) ( ) i meromorphically continuable to, + ( ) (), + j! + j finally Γ() ^ + j ( ) j! + j 2 pole of gamma function

22 x x j = Example x e f( x) =, f ( ) =Γ( ) ζ ( ), + x e x + j e x = Bj +, B =, B =, L e ( j+ )! 2 + j + ^ Γ() ζ () Γ() ^ + j = j = B j ( ) ( j+ )! + j j! + j Bm + ζ() ^, ζ()=, ζ( m) = 2 m+ reult: ζ ( ) i meromorphic in with the only pole at = 22

23 + f( x) = = ( ) + x + / x + / x x + n= f x n= + = ( ) n π () =,, inπ x Example 2 x n n x n ( ) ( ) ^,, (continuation to the left of the f..) + n + n f n= ( ) ( ),, + (continuation to the right of the f..) n + n f ^ n= π ( ) () ^ ( ) inπ x + n n f n 23

24 Example 3 (nonexplicit tranform) f x = = x + x x + x + x x coh x f ( ) O( ) f f ( ) ha a fundamental trip, () ^ + +, () ^ L

25 Convere mapping (/3) Theorem 3 Let ( ) C(, + ) have a tranform ( ) with nonempty fundamental trip αβ,. Let f ( ) be meromorphically continuable to, with a finite number of pole there, and be analytic on R ( ) = γ. Let f ( L f x f ( r ) γβ ) = O with r> when + in γβ,. 25

26 Convere mapping (2/3) Theorem 3 (continue) L f x ^ dξ, γ, β ( ξ ) If ( ) ( ξ, ) A, γ < ξ β Then an aymptotic expanion of f( x) at i f x d ( ) x x x ξ γ ( ) = ξ, (log ) + O( ) ( ξ, ) A ( )! 26

27 Convere mapping (3/3) f () f() x Pole at ξ Term in aymptotic expanion left of f.. expanion at right of f. expanion at + Simple pole ξ left: x at ξ ξ right: x at + ξ Multiple pole Logarifmic factor ( ) ξ left: x (log x) at + ( ξ )! ( ) ξ right: x (log x) at + + ( ξ )! x ξ 27

28 j= ν /2+ i ν /2 i Example 4 Γ() Γ( ν ) φ( ) =, analytic in trip, ν Γ( ν ) + j ( ) Γ ( ν + j ) φ( ) ^, ν j! Γ ( ν ) + j f( x) = φ( ) x d - original function 2π i ( ) Γ ( ν + j ) f x x x x j! Γ( ν ) M j j M+ 2 ( ) = + O( ) j= ν f ( x) = ( + x) ha theame expanion at ν M f( x) = ( + x) + ϖ( x), where ϖ( x) = O( x ) M > 28

29 n= n= Example 5 π φ() =Γ( ) inπ, + n φ() ^ () n! + n, + n n f( x) ( ) n! x - the expanion i only aymptotic! M n n M+ 2 ( ) = ( )! + O( ) f x n x x x n= In fact f( x) = + t e + xt dt 29

30 Definition 3 Harmonic um Gx ( ) = λ g( µ x) harmonic um λ j j j amplitude µ j frequencie Λ( ) = λµ The Dirichlet erie j [ ] µ = µ M g( x); g ( ) G =Λ g () () () j j (eparation property) 3

31 Harmonic um formula The Mellin tranform of the harmonic um Gx ( ) = λ g( µ x) i defined in the interection of the the fundamental trip of g( x) and the domain of abolute convergence of the Dirichlet erie Λ ( ) = λµ (which i of the form R ( ) >σ for ome real σ ) G =Λ( g a () ) () a 3

32 Example x/ x hx ( ) =,,, gx ( ) x = λ = µ = = = + = + x/ + x hn ( ) = Hn = L n () λµ ζ( ) = = Λ = = = π h () =Λ () g () = ζ ( ), inπ ζ() = + γ + L, ζ( ) = + γ + γ ζ ( ) h () ^ 2 ( ), + = ( ) B Hn log n+ γ + log n+ γ L 4 n 2n 2n 2n 32

33 Example 7 + x x lx ( ) = logγ( x+ ) γ x= log + 2, n= n n < + > λ n =, µ n =, gx ( ) = x log( + x) n + + n= n= Λ( ) = λµ = n = ζ( ) π () =Λ( ) () = ζ ( ), 2, inπ l g γ log 2 π + n ( ) ζ ( n) l () ^ ( + ) ( + ) 2 n= n ( + n) + B2 n lx ( ) = log( x!) γx= xlog x ( γ) x log x+ log 2π + n 2 2 n(2n ) x + x x 2n ( π ) log( x!) = log x e 2 x + n= B 2 n(2n ) x 2n Simple pole in poitive integer Double pole in and - n= 2 33

34 n= Example 8 + nx e + n w Lw( x) = (polylogarithm Liw( z) = z n ) w= w n= n 34,, ( ) x λ n = µ w = n g x = e n Λ( = + L = + Γ( + L ) ζ( ), ( ) ζ( ) ), ζ ( n) ( ) H! + ( )! ( + ) + + n () ^ () n= n n n, + + ( ) nζ ( n) n ( )! n= n! n L ( x) = x [ log x+ H ] + ( ) x + π nζ ( n) 2 L2( x) = + ( ) x x n! n= n

35 Example 9 35 modified theta function nx e Θ ( x) =, λ =, µ = n, g( x) = e w w n w n n= n n Θ () = Γ ζ ( + ), double pole at = and imple pole at = 2m γ Θ () ^ L γ 2 4 Θ ( x) = logx+ + x + x + L x Θ () = π M Γ ζ ( ) Θ = + O( x ) x x 2 x 2

36 = Example x Dx ( ) = de ( ), λ = d ( ), µ =, gx ( ) = e d ( ) Λ( ) = = = ( ) + + λµ 2 ζ = = 2 D =Γ( ζ D + () ) () () ^ + 2 γ ζ ( 2 ) ( ) 4 = 2+! ζ ( 2 ) Dx ( ) ( logx+ γ )+ x x 4 2! ( ) = ( + ) 2 + x 36

37 The end 37