The polylogarithm function (or Jonquière's function) of index and argument is a special function, defined in the complex plane for and by analytic continuation otherwise. It can be plotted for complex values ; for example, along the celebrated critical line for Riemann's zeta function [1]. The polylogarithm function appears in the Fermi–Dirac and Bose–Einstein distributions and also in quantum electrodynamics calculations for Feynman diagrams. The 2D plot shows the function , and the 3D plot shows .

For , it is equivalent to the natural logarithm, . For and , it is called the dilogarithm and the trilogarithm; the integral of a polylogarithm is itself a polylogarithm

[2] T. M. Apostol. "Zeta and Related Functions." NIST Digital Library of Mathematical Functions, Version 1.0.9, Release date 2014-08-29. dlmf.nist.gov/25.12.