The Hurwitz zeta function $\zeta_{\mathrm{H}}(s, q)$ is defined for $\operatorname{Re}(q)>0$ by

$$\zeta_{\mathrm{H}}(s, q)=\sum_{n=0}^{\infty} \frac{1}{(q+n)^{s}}$$

State without proof the complex values of $s$ for which this series converges.

Consider the integral

$$I(s, q)=\frac{\Gamma(1-s)}{2 \pi i} \int_{\mathcal{C}} d z \frac{z^{s-1} e^{q z}}{1-e^{z}}$$

where $\mathcal{C}$ is the Hankel contour. Show that $I(s, q)$ provides an analytic continuation of the Hurwitz zeta function for all $s \neq 1$. Include in your account a careful discussion of removable singularities. [Hint: $\Gamma(s) \Gamma(1-s)=\pi / \sin (\pi s)$.]

Show that $I(s, q)$ has a simple pole at $s=1$ and find its residue.