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.
Polylogram heights on the complex plane |
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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.
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Comprehensive analysis and operational calculus
Terms: Comprehensive analysis and operational calculus