Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values

Transcription

1 PolyLog Notations Traditional name Diarithm Traditional notation Li Mathematica StandardForm notation PolyLog, Primary definition Li k k ; k Specific values Specialied values Π p Li q q q k Π p k q Ζ, k q ; p q p q Li exp Π p q q q k exp Π k p q Ψ k q Π 6 q ; p q Values at fixed points Li Li Π

2 Li Ν Π Li Π (L.Euler) Li Π Π Li C Π Li C Π Li Π Π C Li Π Π C 6 4 Li 5 Li Π 5 Π 0 Li 3 5 Li Π Π 0 5 Li Π 5 Π C 8 96 General characteristics

3 3 Domain and analyticity Li is an analytical function of which is defined in Li Symmetries and periodicities Mirror symmetry Li Li ;, 0 Periodicity No periodicity Poles and essential singularities The function Li does not have poles and essential singularities ing Li Ν Branch points The function Li has two branch points:, Li Ν, Li, Branch cuts The function Li is a single-valued function on the -plane cut along the interval,, where it is continuous from below Li,, lim Li x Ε Li x ; x Ε lim Li x Ε Li x Π x ; x Ε0 Series representations Generalied power series

4 4 Expansions at 0 For the function itself Li ; Li O4 Li k k ; k Li 3 F,, ;, ; Li O Li F ; m F m Li m,, m m k k Summed form of the truncated series expansion. Expansions at For the function itself k Li Π 3 Log ; Li Π 3 O 4 Log O Li Π 6 k k k k k k ; (L.Euler, 768) Li Li Π 6 (L.Euler, 768)

5 Li Π 6 Ζ j j j j Li Π F,, ;, ; F, ; ; Li Π O O Li F ; m k F m k k,, m m Π Summed form of the truncated series expansion. Expansions at For the function itself Π m 6 k k k 6 m, 0 Li m Li Π Li Π 6 ; 4 4 O Li Π 6 k ; k k Li Π 6 Li n Li Π 6 O ; 0, Li F ; F n Π n 6 k k k Π 6,, n n Li n Summed form of the truncated series expansion.

6 6 For the function itself OTHER TITLE? Li k k k k Π 6 ; Residue representations s 3 s Li res s s j ; s j Li res s s s s j ; j s Integral representations On the real axis Of the direct function t Li 0 t t Li t t t t t Li t ; 0 t t t k Li 0 t k k k t ; Re t Li t 0 t Li Π 6 t t t Contour integral representations

7 Li Π s s3s s ; arg Π s Li Γ Π s s s s ; Γ 0 arg Π Γ s Multiple integral representations w Li 0 w t w t Limit representations F ε, ε; ; Li lim ε0 ε Differential equations Ordinary linear differential equations and Wronskians For the direct function itself w w ; w c c Li w 3 3 w w 0 ; w c c c 3 Li W,, Li w w w w 0 ; w c c c 3 c 4 Li W,,, Li k d d d d l d d w 0 ; w Li There is no algebraic partial differential equation for Li Ν (A. Ostrowski, 90).

8 w 3 3 g g g g 3 g g w c c g c 3 Li g w g 3 W, g, Li g g g g g g 3 g g g g g 3 g g g3 g w 0 ; w 3 3 g g g g 3 h h 3 g g g h g 3 g h g g h w 6 h 6 g h 3 g h h g g g g g 3 g g 3 h h g3 g w 6 h 3 6 g h 4 g h 6 g h 6 h h 3 g h h 3 g h g g h h g h h g g h h g g g g h 0 ; w c h c h g c 3 h Li g 3 h g g h g h 3 g h h g 3 h g h3 h w W h, h g, h Li g h3 g 3 3 w g g r a r 3 s 3 w 3 s 3 s r r s a r w s a s r r s w 0 ; a r w c s c s a r c 3 s Li a r W s, s a r, s Li a r a r3 r3 s3 a r w 3 r a r 3 s w 3 s r a r sw r a r s sw 0 ; w c s c s a r c 3 s Li a r W s, s a r, s Li a r a r s 3 3 r a r Transformations Multiple arguments

9 Li w Li w Li Li w w Li w w w w w w w w ; (L.Rogers, 906) Li w Li Li w Li w w Li w w ; w w w w w w w w (C.J.Hill,830) w Li w Li Li w Li w w w Li w w w w w w w w w w w ; 0 w w 0 w 0 w 0 w w Power of arguments Li Li Li Li m m m k 0 Π k Li m ; m Identities Functional identities Involving two diarithms Li Li Li Li Π 6 (L.Euler, 768) Li Li Π 6 ; 0,

10 Li Li Π Π Li Π Π 3 Li Li Li Π Π Li Li Π 6 ; Re 0, Li Li Π Li Li ;, (J.Landen,780) Li Li Π Π ; Li Li Π Π Π ; Involving three diarithms Li Li Li ; 0, Li Li Li Abel's duplication formula ; Im Li Π 6 Li Li ; Li Li Li 4 ; Re 0 Im 0

11 Li Li Li 4 ; Re 0 Im Li Li Li Li Li Li Re 0 Re 0 Im 0 ; Involving four diarithms Li Li Li Li Π 4 ;,, Involving five diarithms L Lw L w L w w L w w 0 ; L Li ; 0 w w 0 0 w w 0 w 0 w 0 w w Li w Li w Li w Li Li w w w w w w w w w w w ; w 0 w w 0 0 w w 0 w w 0 w w w Li Li w w Li Li w Li w w w 0 w (W.Spence, 809) w w ; w Li Li w w Li w Li Li w w ; w w w (N.Abel, 830)

12 Li w Li w Li Li w w Li w w w w w w w w ; (L.Rogers, 906) Li w Li Li w Li w w Li w w ; w w w w w w w w (C.J.Hill,830) Li w Li Li w Li w w w Li w w w w w w w w w w ; w w 0 w w 0 w 0 w 0 w w Li Li w Li w w Li w w Li w w ; w Li w w Li w Li Li w Li w w w 0 w 0 w w w 0 (W.Schaeffer, 846) Π 6 ; w w Li Li w w 0 w w w Li Li w w Li w w w ; (E.Kummer, 840) Involving six diarithms Li x Li y Li Li x y Li x Li y ; x y x y x y Li x Li y Li Li y x x y Li x x Li y y ; x y y x x y x y

13 Li x Li y Li Li y x Li x y Li x y ; y x y x Newman's formula Li x Li y Li Li x y Li x Li y ; x y x y x y Involving nine diarithms v w Li x y Li v x Li w x Li v y Li w y Li x Li y Li v Li w x y ; v w x y 0 x 0 y 0 v 0 w (W.Mantel, 898) Involving several diarithms n n Li Li k Λ p Li k p Λ k Λ p Π n 6 ; p Λ p k k n Relations of special kind f w f w f w w f w w Π f f f 6 ; f Li 0 Li is the unique solution of class C3 0, of the functional equations f w f w f w w for all real 0 0 w. w Π f f f f w 6 Complex characteristics Real part ReLi x y 4 x y x x y x y x y y x x y Li x y x y Li x y x y Differentiation Low-order differentiation

14 Li Li Symbolic differentiation m Li m m m j 0 Sm j m Li j k k k j ; n m Li k m k m k 0 k m ; n m Li m m Sm j, j, m ; n j m Li m m m, m ; m m m Li m F m, m; m ; ; m m Fractional integro-differentiation Α Li Α Α Α 3 F,, ;, Α; Α Li k kα ; Α k Α k k Integration Indefinite integration Involving only one direct function Li Li

15 Li k k ; k k Involving one direct function and elementary functions Involving power function Α Li k Α Li Α kα ; k Α k Α 4F 3 Α,,, ; Α,, ; Li Ν a Li Ν a Involving rational functions Li Li Li Li Li Li Involving exponential function Li Li 3 Involving rational functions and arithm Li Li 3 Li Li 3 3 Li Li 6 Li 3 6 Li 4 Definite integration For the direct function itself

16 t Li tt Li t Α Li tt Α Π Α 6 Α Α ΨΑ 6 6 Α Α 3 ; ReΑ t n Li tt Π 0 6 n n ; n n k k t 3 Li t t 8Π t 5 Li t t 8Π Π 7 8 Π 0 Li t t Li t Π ;, Li t t 4 Π Li t t Li tan tt Π Li Involving the direct function t a t Li t t 480 a 0 Π, 3, a a 5 4 a 50 Π a 0 Li 3 a a 53 Π Li 4 a t 34 t Li t For the products of direct functions t Π 6 3 C 4 5 Π 3 0 t Li t Li t t 6 Π t 34 Li t Li t 56 Π 0 t 3 C 3

17 t Li a t Li t t 880 a 360 Π, 4, a a 3 5 a 50 Π 3 a 59 Π 4 a 360 Li 4 a a 440 Li 5 a t Li nt Li t n Ζn 3 ; n t Representations through more general functions Through hypergeometric functions of two variables Li F ;, ;, ;, 3;;; Li Li w Li w w w F Li 0 F 0 0 ;, ;, ;, 3;;; ;, ;, ;, w 3;;; Li F 0 0 ;, ;, ; 3;;;, Π Through hypergeometric functions Involving p F q Li 3 F,, ;, ; Through Meijer G Classical cases for the direct function itself Li G,3 3,3,,, 0, 0 Through other functions Li S, Li,,

18 8 Representations through equivalent functions With related functions Π p Li q q q k Π p k q Ζ, k q ; p q p q Theorems The Fermi Dirac integral The Fermi-Dirac integral F Α Μ 0 ε Α eμ ε, describing, for instance, the number of electrons (holes) in the conduction band (valence band) in a semiconductor with density of states ε Α, can be expressed as F Α Μ Α Α Li Α Μ. The volume of a Lambert cube The volume V of a Lambert cube with essential angles Α, Α, Α 3 and apices of length l, l, l 3 in three-dimensional hyperbolic space is given by 3 V Α k Θ Α k Θ 4Θ Π Θ ; ImLi exp k tan coshl sin Α sin Α cos Α cos Α. Rationality of diarithm The value Li is irrational when is rational (G.V.Chudnovsky,979). History G. W. Leibni defined diarithm for the case Ν L. Euler (768) J. Landen (760,780) inverstigated Li and Li 3 W. Spence (809) N. H. Abel E.E. Kummer (840) J. Kummer; L. L. Lindelöf N. I. Lobachevski C. J. Hill (88) introduced the name "diarithm" Applications include electrical network problems, number theory, group theory, K-theory, geometry, quantum electrodynamics, group cohomoy, mixed Hodge structures, mixed motives, evaluation of volumes of hyperbolic polytopes, celestial mechanics.

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