Abstract

This entry provides a definition of the Polylogarithm function, commonly denoted as Li(a,z). Here, z is a complex number and a an integer parameter. This function can be defined by a power series for |z| < 1 and analytically extended to the entire complex plane, except for a branch cut on the reals ≥ 1.

Several basic properties are also proven, such as the relationship to the Eulerian polynomials, the derivative formula, the relationship to the ‘normal’ logarithm, and the duplication formula.

BibTeX

@article{Polylog-AFP,
  author  = {Manuel Eberl},
  title   = {The Polylogarithm Function},
  journal = {Archive of Formal Proofs},
  month   = {November},
  year    = {2023},
  note    = {\url{https://isa-afp.org/entries/Polylog.html},
             Formal proof development},
  ISSN    = {2150-914x},
}