The Hankel representation of the gamma function is

$$\Gamma(z)=\frac{1}{2 i \sin (\pi z)} \int_{-\infty}^{\left(0^{+}\right)} t^{z-1} e^{t} d t$$

where the path of integration is the Hankel contour.

Use this representation to find the residue of $\Gamma(z)$ at $z=-n$, where $n$ is a nonnegative integer.

Is there a pole at $z=n$, where $n$ is a positive integer? Justify your answer carefully, working only from the above representation of $\Gamma(z)$.