The Lerch $\Phi$ Analytic Continuation

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.