PolyLog

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed ν

Values at fixed points

Dilogarithmical case

Trilogarithmical case

General characteristics

Domain and analyticity

(Subscript[Li, ν])(z) is an analytical function of ν, z which is defined in (ℂ^2).

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

For fixed ν, the function (Subscript[Li, ν])(z) does not have poles and essential singularities.

With respect to ν

For fixed z, the function (Subscript[Li, ν])(z) has only one singular point at ν = (OverTilde[∞]).  It is an essential singular point.

Branch points

With respect to z

With respect to ν

For fixed z, the function (Subscript[Li, ν])(z) does not have branch points.

Branch cuts

With respect to z

With respect to ν

For fixed z, the function (Subscript[Li, ν])(z) does not have branch cuts.

Series representations

Generalized power series

Expansions a z==0

Expansions a z==1

Expansions at z==∞

Residue representations

Allan Cortzen

Other series representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Multiple integral representations

Limit representations

Differential equations

Ordinary linear differential equations and Wronskians

For the direct function itself

There is no algebraic partial differential equation for (Subscript[Li, ν])(z) (A. Ostrowski, 1920).

Transformations

Multiple arguments

Dilogarithmical case

Power of arguments

Identities

Recurrence identities

General cases

Functional identities

General cases

Dilogarithmical cases

Trilogarithmical cases

Relations of special kind

Complex characteristics

Real part

Imaginary part

Differentiation

Low-order differentiation

With respect to ν

With respect to z

Symbolic differentiation

With respect to ν

With respect to z

Fractional integro-differentiation

With respect to ν

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving only one direct function with respect to ν

Involving one direct function and elementary functions with respect to ν

Definite integration

For the direct function itself

Involving the direct function

For the products of direct functions

Summation

Infinite summation

Representations through more general functions

Through hypergeometric functions of two variables

Through hypergeometric functions

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Through other functions

Representations through equivalent functions

With related functions

Inequalities

Theorems

The Fermi–Dirac integral

The Fermi-Dirac integral (Subscript[F, α])(μ)==(∫_0^∞)((ε^α)/((e^(e-μ))+1))dε , 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 (Subscript[F, α])(μ)==-α Γ(α) (Subscript[Li, α+1])(-(e^μ)).

The volume of a Lambert cube

The volume V of a Lambert cube with essential angles (Subscript[α, 1]),  (Subscript[α, 2]), (Subscript[α, 3]) and apices of length  (Subscript[l, 1]),  (Subscript[l, 2]), (Subscript[l, 3]) in three-dimensional hyperbolic space is given by

V==(∑_(k=1)^3)(ℒ((Subscript[α, k])+θ)-ℒ((Subscript[α, k])-θ))-1/4ℒ(2θ)+1/2ℒ(π/2-θ)/;ℒ(z)==1/2 Im((Subscript[Li, 2])(exp(2 i z)))∧
					(tan^(-1))((cosh((Subscript[l, 2]))-(sin^2)((Subscript[α, 1]))(sin^2)((Subscript[α, 2])))/((cos^2)((Subscript[α, 1]))(cos^2)((Subscript[α, 2])))).

Rationality of dilogarithm

The value (Subscript[Li, 2])(z) is irrational when zis rational (G.V.Chudnovsky,1979).

History

– G. W. Leibniz defined dilogarithm for the case ν=2 
– L. Euler (1768)
– J. Landen (1760,1780) inverstigated (Subscript[Li, 2])(z) and (Subscript[Li, 3])(z)
– W. Spence  (1809) 
– N. H. Abel
– E.E. Kummer  (1840) 
– J. Kummer; L. L. Lindelöf
– N. I. Lobachevski
– C. J. Hill (1828) introduced the name "dilogarithm"

Applications include electrical network problems, number theory, group theory, K-theory, geometry, quantum electrodynamics, group cohomology, mixed Hodge structures, mixed motives, evaluation of volumes of hyperbolic polytopes, celestial mechanics.

Copyright