Consider the integral

$$\int_{0}^{\infty} \frac{t^{z} \mathrm{e}^{-a t}}{1+t} d t$$

where $t^{z}$ is the principal branch and $a$ is a positive constant. State the region of the complex $z$-plane in which the integral defines a holomorphic function.

Show how the analytic continuation of this function can be obtained by means of an alternative integral representation using the Hankel contour.

Hence show that the analytic continuation is holomorphic except for simple poles at $z=-1,-2, \ldots$, and that the residue at $z=-n$ is

$$(-1)^{n-1} \sum_{r=0}^{n-1} \frac{a^{r}}{r !}$$

Part II