Built-in Mathematica Symbol

PolyLog

PolyLog[n, z]
gives the polylogarithm function

.

PolyLog[n, p, z]
gives the Nielsen generalized polylogarithm function

.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

.

.

.

PolyLog[n, z] has a branch cut discontinuity in the complex plane running from to .

For certain special arguments, PolyLog automatically evaluates to exact values.

PolyLog can be evaluated to arbitrary numerical precision.

PolyLog automatically threads over lists.

Scope (7)

Simple exact values are generated automatically:

Evaluate with large order and argument:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex order and arguments:

PolyLog threads element-wise over lists:

TraditionalForm formatting:

Generalizations & Extensions (7)

Infinite arguments give symbolic results:

PolyLog can be applied to power series:

Evaluate derivatives exactly:

Series expansion at branch cuts:

Series expansion at infinity:

Give the result for an arbitrary symbolic direction:

Special cases:

Series expansion:

Applications (4)

Plot of the absolute value of the dilogarithm function in the complex plane:

Calculate integrals over Fermi-Dirac distributions:

Volume of a hyperbolic dodecahedron with vertices 0, 1,

,

(subject to Im(z)>0):

Plot the volume as a function of the vertex

:

Mahler measure of the trivariate polynomial

as a function of

:

Plot the Mahler measure:

Properties & Relations (6)

Use FullSimplify to simplify polylogarithms:

Use FunctionExpand to expand polylogarithms:

Numerically find a root of a transcendental equation:

Integration:

Generate from integrals and sums:

PolyLog appears in special cases of various mathematical functions:

Possible Issues (1)

Large orders can give results too large to be computed explicitly: