Difference between revisions of "Contour integral representation of reciprocal gamma"

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Revision as of 00:35, 24 May 2016

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: