Polylogarithm

Polylogarithm is a special function denoted by and defined as an infinite power series

where s and z are complex numbers, and . For other z, a generalization is made using analytic continuation.

A special case is in which . Functions and were named dilogarithm and trilogarithm respectively. For polylogarithms of various orders, the ratio

Alternative definitions of the polylogarithm are Fermi-Dirac and Bose-Einstein integrals.

Partial Values

Literature

• Abel NH compuvres complètes de Niels Henrik Abel - Nouvelle édition, Tome II. - Christiania [Oslo]: Grøndahl & Søn, 1881. - P. 189–193. (this 1826 manuscript was only published posthumously.)
• Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. - New York: Dover Publications, 1972. - ISBN 0-486-61272-4.
• (April 1997) "On the Rapid Computation of Various Polylogarithmic Constants" (PDF). Mathematics of Computation 66 (218): 903–913. DOI: 10.1090 / S0025-5718-97-00856-9.