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PolyLog

gives the polylogarithm function .

PolyLog[n,p,z]

gives the Nielsen generalized polylogarithm function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
$TemplateBox[{n, z}, PolyLog]=sum_(k=1)^(infty)z^k/k^n$ .
$TemplateBox[{n, p, z}, PolyLog3]=(-1)^(n+p-1)/((n-1)!p!)int_0^1log^(n-1)(t)log^p(1-zt)/t dt$ .
.
PolyLog[n,z] has a branch cut discontinuity in the complex plane running from 1 to .
For certain special arguments, PolyLog automatically evaluates to exact values.
PolyLog can be evaluated to arbitrary numerical precision.
PolyLog automatically threads over lists.
PolyLog can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples (6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

Scope (33)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix PolyLog function using MatrixFunction:

Specific Values (5)

Simple exact values are generated automatically:

PolyLog for symbolic z:

PolyLog for symbolic n:

Value at zero:

Find a value of z for which PolyLog[1,z ]=1:

Visualization (3)

Plot the PolyLog function as a function of its parameter n:

Plot the PolyLog function for various orders:

Plot the real part of PolyLog function:

Plot the imaginary part of PolyLog function:

Function Properties (11)

Real domain of PolyLog:

Complex domain:

Function range of :

PolyLog threads elementwise over lists:

PolyLog is not an analytic function:

PolyLog is not meromorphic:

is non-decreasing on its real domain for :

For other values of , it might or might not be monotonic:

is injective for :

is not surjective for :

PolyLog is neither non-negative nor non-positive:

PolyLog has both singularity and discontinuity for x≥1:

is convex on its real domain:

TraditionalForm formatting:

Differentiation (2)

First derivatives with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when n=1/2:

Series Expansions (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Function Identities and Simplifications (4)

PolyLog is defined through the identity:

Recurrence identity:

For positive integer , can be expressed in terms of hypergeometric functions:

For negative integer , is a rational function of :

Generalizations & Extensions (7)

Ordinary Polylogarithm Function (5)

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:

Nielsen Generalized Polylogarithm Function (2)

Special cases:

Series expansion:

Applications (5)

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

Calculate integrals over Bose–Einstein distributions:

Calculate integrals over Fermi–Dirac distributions:

Volume of a hyperbolic ideal tetrahedron with vertices at , 0, 1, (subject to ):

Plot the volume as a function of the vertex :

Mahler measure of the trivariate polynomial as a function of :

Plot the Mahler measure:

Generate the Eulerian numbers [MathWorld]:

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:

Neat Examples (1)

Plot the Riemann surface of the dilogarithm :

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