The Polylogarithm Function

Manuel Eberl πŸ“§

November 15, 2023

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.

Abstract

This entry provides a definition of the Polylogarithm function, commonly denoted as Lis(z). Here, z is a complex number and s an integer parameter. This function can be defined by the power series expression Lis(z)=βˆ‘k=1∞zkks for |z|<1 and analytically extended to the entire complex plane, except for a branch cut on Rβ‰₯1.

Several basic properties are also proven, such as the relationship to the Eulerian polynomials via Liβˆ’k(z)=z(1βˆ’z)kβˆ’1Ak(z) for kβ‰₯0, the derivative formula ddzLis(z)=1zLisβˆ’1(z), the relation to the β€œnormal” logarithm via Li1(z)=βˆ’ln⁑(1βˆ’z), and the duplication formula Lis(z)+Lis(βˆ’z)=21βˆ’sLis(z2).

License

BSD License

Topics

Session Polylog

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