# PolyLog

PolyLog[n,z]

gives the polylogarithm function .

PolyLog[n,p,z]

gives the Nielsen generalized polylogarithm function .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• .
• .
• 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 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 x1:

is convex on its real domain:

### 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 BoseEinstein distributions:

Calculate integrals over FermiDirac 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 :

Wolfram Research (1988), PolyLog, Wolfram Language function, https://reference.wolfram.com/language/ref/PolyLog.html (updated 2022).

#### Text

Wolfram Research (1988), PolyLog, Wolfram Language function, https://reference.wolfram.com/language/ref/PolyLog.html (updated 2022).

#### CMS

Wolfram Language. 1988. "PolyLog." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/PolyLog.html.

#### APA

Wolfram Language. (1988). PolyLog. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolyLog.html

#### BibTeX

@misc{reference.wolfram_2024_polylog, author="Wolfram Research", title="{PolyLog}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/PolyLog.html}", note=[Accessed: 07-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_polylog, organization={Wolfram Research}, title={PolyLog}, year={2022}, url={https://reference.wolfram.com/language/ref/PolyLog.html}, note=[Accessed: 07-August-2024 ]}