gives the polylogarithm function TemplateBox[{n, z}, PolyLog].


gives the Nielsen generalized polylogarithm function TemplateBox[{n, p, z}, PolyLog3].


  • 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.
  • TemplateBox[{{n, -, 1}, 1, z}, PolyLog3]=TemplateBox[{n, z}, PolyLog].
  • 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. »


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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  (32)

Numerical Evaluation  (5)

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:

PolyLog can be used with Interval and CenteredInterval objects:

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 TemplateBox[{2, x}, PolyLog]:

PolyLog threads elementwise over lists:

PolyLog is not an analytic function:

PolyLog is not meromorphic:

TemplateBox[{n, x}, PolyLog] is non-decreasing on its real domain for :

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

TemplateBox[{n, x}, PolyLog] is injective for :

TemplateBox[{n, x}, PolyLog] is not surjective for :

PolyLog is neither non-negative nor non-positive:

PolyLog has both singularity and discontinuity for x1:

TemplateBox[{2, x}, PolyLog] 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 , TemplateBox[{n, z}, PolyLog] can be expressed in terms of hypergeometric functions:

For negative integer , TemplateBox[{n, z}, PolyLog] 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:


Generate from integrals and sums:

PolyLog appears in special cases of various mathematical functions:

Neat Examples  (1)

Plot the Riemann surface of the dilogarithm TemplateBox[{2, z}, PolyLog]:

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


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


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


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


@misc{reference.wolfram_2023_polylog, author="Wolfram Research", title="{PolyLog}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/PolyLog.html}", note=[Accessed: 12-April-2024 ]}


@online{reference.wolfram_2023_polylog, organization={Wolfram Research}, title={PolyLog}, year={2022}, url={https://reference.wolfram.com/language/ref/PolyLog.html}, note=[Accessed: 12-April-2024 ]}