So I want to prove that for all integers $n \in \mathbb{Z}$ it holds that $$F(n):= \frac{1}{2\pi i} \int\limits_{\gamma}^{} s^{-n}e^{s} ds = \frac{1}{\Gamma(n)},$$ with $\gamma$ the 'Hankel'-contour:
I've already proven the more general case, i.e. that $$F(z)= \frac{1}{2\pi i} \int\limits_{\gamma}^{} s^{-z}e^{s} ds = \frac{1}{\Gamma(z)},$$ for all $z \in \mathbb{C},$ but I also want to prove this special case separately.
Just to be complete: the gamma function is defined as $\int\limits_{0}^{\infty} t^{z-1}e^{-t}dt$ and has the property that for all $n=1,2,\ldots$ it holds that that $\Gamma(n) = (n-1)!$.
Any help would be dearly appreciated.
EDIT: I now have successfully proven that $$F(z-1)=(z-1) \cdot F(z),$$ and for all $n \in \mathbb{N}$ I've proven the property. But now I am stuck on the question for all negative numbers.