 Solution

For $|z|<1$ on the negative-real portions of the Hankel contour, expandLocal uniform convergence permits termwise integration. The reciprocal Hankel formula givesso $I(z,s)={\sum }_{n\ge 1}{z}^n/n^s={\mathrm{Li}}_s(z)$ initially on the disc.

For $z\notin [1,\infty )$, choose the contour around the negative axis so that it passes between zero and every pole satisfying $e^{-t}=z$. On compact subsets of the slit $z$-plane it can be chosen uniformly, and differentiation under the integral proves holomorphy. Deforming without crossing a pole gives a single-valued analytic continuation. Near the cut, the pole $t=-\log z$ lies on opposite sides of the two admissible contours; this is why the contour must pass it consistently.