Difference between revisions of "Contour integral representation of reciprocal gamma"
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| − | <strong>Theorem:</strong> The following formula holds for a [[positively orientation|positively oriented]] [[contour]] $C$ is a path encircling $0$ beginning at and returning to $+\infty$: | + | <strong>[[Contour integral representation of reciprocal gamma|Theorem]]:</strong> The following formula holds for a [[positively orientation|positively oriented]] [[contour]] $C$ is a path encircling $0$ beginning at and returning to $+\infty$: |
$$\dfrac{1}{\Gamma(z)} = \dfrac{i}{2\pi} \displaystyle\oint_C (-t)^{-z}e^{-t} \mathrm{d}t,$$ | $$\dfrac{1}{\Gamma(z)} = \dfrac{i}{2\pi} \displaystyle\oint_C (-t)^{-z}e^{-t} \mathrm{d}t,$$ | ||
where $\dfrac{1}{\Gamma}$ denotes the [[reciprocal gamma]] function, $\pi$ denotes [[pi]], and $e^{-t}$ denotes the [[exponential]] function. | where $\dfrac{1}{\Gamma}$ denotes the [[reciprocal gamma]] function, $\pi$ denotes [[pi]], and $e^{-t}$ denotes the [[exponential]] function. | ||
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: █
