The Lerch $\hat{I}¦$ Analytic Continuation

Abstract

We demonstrate how to extend formulae for the Lerch transcendent function, $\hat{I}¦(e^z,k,b)$, 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.

Formulae were fixed due to multi-valued functions, which can't be simplified. Notation and writing were improved, and some inaccurate statements were rectified