The Lerch $\Phi$ Analytic Continuation

TL;DR

This paper develops methods to extend the formulas for the Lerch transcendent and polylogarithm functions from positive integer parameters to the entire right half of the complex plane, enhancing their analytical scope.

Contribution

It introduces a technique for the analytic continuation of the Lerch transcendent and polylogarithm functions beyond positive integers, expanding their applicability.

Findings

Extended formulas valid for Re(k)>0

Analytic continuation of partial sums

Broadened domain for special functions

Abstract

We demonstrate how to extend formulae for the Lerch transcendent function, $\Phi(e^z,k,b)$, and the polylogarithm, $\mathrm{Li}_{k}(e^{z})$, that only hold at the positive integers to the right half of the complex $k$-plane, that is, $\Re{(k)}>0$. The same is done for the partial sums of each of these functions.