gives the polylogarithm function .
gives the Nielsen generalized polylogarithm function .
- 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. »
Examplesopen allclose all
Basic Examples (6)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Numerical Evaluation (5)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
PolyLog can be used with Interval and CenteredInterval objects:
Specific Values (5)
Function Properties (11)
Real domain of PolyLog:
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:
PolyLog is neither non-negative nor non-positive:
PolyLog has both singularity and discontinuity for x≥1:
Series Expansions (2)
Find the Taylor expansion using Series:
Plots of the first three approximations around :
Function Identities and Simplifications (2)
PolyLog is defined through the identity:
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)
Plot of the absolute value of the dilogarithm function in the complex plane:
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:
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:
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