Difference between revisions of "Reciprocal gamma"
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{{:Contour integral representation of reciprocal gamma}} | {{:Contour integral representation of reciprocal gamma}} | ||
Revision as of 09:41, 4 June 2016
The reciprocal gamma function is the function $\dfrac{1}{\Gamma(z)}$, where $\Gamma$ denotes the gamma function.
Graph of $\dfrac{1}{\Gamma}$ on $[-4,10]$.
Graph of $\dfrac{1}{\Gamma}$ on $[-7.5,5.1]$.
Domain coloring of $\dfrac{1}{\Gamma}$.
Plot of $\Gamma$ and $\dfrac{1}{\Gamma}$ from Abramowitz&Stegun.
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Properties
Reciprocal gamma written as an infinite product Gamma function Weierstrass product
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.
