PolyLog Function

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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
.
Contributed by: Enrique Zeleny (November 2014)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The polylogarithm function is defined as
.
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
.
References
[1] L. Vepstas. "Polylogarithm, The Movie." (Nov 20, 2014) linas.org/art-gallery/polylog/polylog.html.
[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.
[3] Souichiro-Ikebe. "Polylogarithm Function." (Dec 4, 2015) Graphics Library of Special Functions (in Japanese). http://math-functions-1.watson.jp/sub1_spec_040.html.
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