Contour integral representation of reciprocal gamma

Theorem: The following formula holds for a positively oriented contour $C$ is a path encircling $0$ beginning at and returning to $+\infty$: $$\dfrac{1}{\Gamma(z)} = \dfrac{i}{2\pi} \displaystyle\oint_C (-t)^{-z}e^{-t} \mathrm{d}t,$$ where $\dfrac{1}{\Gamma}$ denotes the reciprocal gamma function, $\pi$ denotes pi, and $e^{-t}$ denotes the exponential function.

Proof: █