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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 |last=Borwein |first= 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(x)=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
|jstor=41505586 |s2cid=85549521 |access-date=20 March 2023}}</ref> (where <math>\Psi(x)</math> is the [[digamma function]]).
[[File:Inverse Gamma Function.png|thumb]]
==== Definition ====
The inverse gamma function may be defined by the following integral representation <ref>{{cite journal |last1=Pedersen |first1=Henrik |title="Inverses of gamma functions" |journal=Constructive Approximation |date=9 September 2013 |volume=7 |doi=10.1007/s00365-014-9239-1 |url=https://link.springer.com/article/10.1007/s00365-014-9239-1}}</ref>
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