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{{Short description|Inverse of the gamma function}}
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{{Distinguish|Inverse-gamma distribution|Reciprocal gamma function}}
In [[mathematics]], the inverse gamma function <math>\Gamma^{-1}(x)</math> is the [[inverse function]] of the [[gamma function]]. In other words, it is the function satisfying <math display="inline">\Gamma(y)=x</math>. For example, <math>\Gamma^{-1}(24)=5</math> <ref>{{Cite journal |last1=Borwein |first1= Jonathan M. |last2=Corless |first2= Robert M.|title=Gamma and Factorial in the Monthly |journal=The American Mathematical Monthly 125.5 |year=2017 |arxiv=1703.05349 }}</ref>. Usually, the inverse gamma function refers to the principal branch on the interval <math>\left(\Gamma(\alpha)= 0.8856031..., \infty\right)</math> where <math>\alpha=1.4616321...</math> is the unique positive number such that <math>\psi(\alpha)=0</math> <ref>{{cite journal |last1=Uchiyama |first1=Mitsuru |title=The principal inverse of the gamma function |date=April 2012 |url= https://www.jstor.org/stable/41505586 |journal=Proceedings of the American Mathematical Society|volume=140 |issue=4 |pages=1347 |doi= 10.1090/S0002-9939-2011-11023-2
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[[File:Inverse Gamma Function.png|thumb]]
==== Definition ====
The inverse gamma function may be defined by the following integral representation
<math>\Gamma^{-1}(x)=a+bx+\int_{-\infty}^{\Gamma(\alpha)}\left(\frac{1}{x-t}-\frac{t}{t^{2}-1}\right)d\mu(t) </math>
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==== Approximation ====
To compute the branches of the inverse gamma function one can first compute the Taylor series of <math>\Gamma(x)</math> near <math>\alpha</math>. The series can then be truncated and inverted, which yields successively better approximations to <math>\Gamma^{-1}(x)</math>. For instance, we have the quadratic approximation
<math>
\Gamma^{-1}\left(x\right)\approx\alpha+\sqrt{\frac{2\left(x-\Gamma\left(\alpha\right)\right)}{\Psi\left(1,\ \alpha\right)\Gamma\left(\alpha\right)}}.</math>
The inverse gamma function also has the following [[asymptotic formula]]
<math>\Gamma^{-1}(x)\sim\frac{1}{2}+\frac{\ln\left(\frac{x}{\sqrt{2\pi}}\right)}{W_{0}\left(e^{-1}\ln\left(\frac{x}{\sqrt{2\pi}}\right)\right)}</math>
Where <math>W_0(x)</math> is the [[Lambert W function]]. The formula is found by inverting the [[Stirling's approximation|Stirling approximation]], and so can also be expanded into an asymptotic series.
'''Series Expansion'''
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To obtain a series expansion of the inverse gamma function one can first compute the series expansion of the [[reciprocal gamma function]] <math>\frac{1}{\Gamma(x)}</math> near the poles at the negative integers, and then invert the series.
Setting <math>z=\frac{1}{x}</math> then yields, for the ''n'' th branch <math>\Gamma_{n}^{-1}(z)</math> of the inverse gamma function (<math>n\ge 0</math>) <ref>{{Cite journal |last1=Couto |first1=Ana Carolina Camargos |last2=Jeffrey |first2=David |last3=Corless |first3=Robert |date=November 2020 |title=The Inverse Gamma Function and its Numerical Evaluation |url=https://www.maplesoft.com/mapleconference/2020/highlights.aspx |at=Section 8 |journal=Maple Conference Proceedings}}</ref>
<math>\Gamma_{n}^{-1}(z)=-n+\frac{\left(-1\right)^{n}}{n!z}+\frac{\psi^{(0)}\left(n+1\right)}{\left(n!z\right)^2}+\frac{\left(-1\right)^{n}\left(\pi^{2}+9\psi^{(0)}\left(n+1\right)^{2}-3\psi^{(1)}\left(n+1\right)\right)}{6\left(n!z\right)^3}+O\left(\frac{1}{z^{4}}\right)</math>
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== References ==
{{reflist}}
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