Difference between revisions of "Contour integral representation of reciprocal gamma"
From specialfunctionswiki
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| − | + | ==Theorem== | |
| − | + | 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. | ||
| − | + | ||
| − | + | ==Proof== | |
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| − | + | ==References== | |
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 07:17, 16 June 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.
