PolyLog3

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed ν,p

For fixed ν, z

For fixed p,z

For fixed ν

For fixed z

Values at fixed points

General characteristics

Domain and analyticity

(Subscript[S, ν,p])(z) is an analytical function of ν, p, z which is defined in (ℂ^3).

Symmetries and periodicities

Mirror symmetry

Permutation symmetry

Permutation symmetry for z==1:

Periodicity

No periodicity

Poles and essential singularities

With respect to z

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

With respect to ν

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

Branch points

With respect to z

For fixed ν, p the function (Subscript[S, ν,p])(z) has two branch points: z==1, z==(OverTilde[∞]).

With respect to ν

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

Branch cuts

With respect to z

With respect to ν

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

Series representations

Generalized power series

Expansions a z==0

Other series representations

Integral representations

On the real axis

Of the direct function

Differential equations

Ordinary nonlinear differential equations

Identities

Functional identities

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

History

– N. Nielsen (1909)
– K. S. Kölbig
– J. A. Mignaco
– E. Remiddi (1970)

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.

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