Inverse gamma function: Difference between revisions

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{{Distinguish|Inverse-gamma distribution|Reciprocal gamma function}}

{{Orphan|date=July 2023}}

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[[File: | image1 = Inverse Gamma Function.png|thumb]]

| caption1 = Graph of an inverse gamma function

| image2 = Inverse gamma function in complex plane.png

[[File:Inverse | gammacaption2 function= in complex plane.png|Plot of inverse gamma function in the complex plane |thumb]]

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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, <math>y = \Gamma^{-1}(x)</math> whenever <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 |year=2017 |volume=125 |issue=5 |pages= 400–424 |doi= 10.1080/00029890.2018.1420983 |arxiv=1703.05349 |jstor=48663320 |s2cid=119324101}}</ref> Usually, the inverse gamma function refers to the principal branch with ___domain on the real interval <math>\left[\beta, +\infty\right)</math> and image on the real interval <math>\left[\alpha, +\infty\right)</math>, where <math>\beta = 0.8856031\ldots</math><ref>{{oeis|A030171}}</ref> is the minimum value of the gamma function on the positive real axis and <math>\alpha = \Gamma^{-1}(\beta) = 1.4616321\ldots</math><ref>{{oeis|A030169}}</ref> is the ___location of that minimum.<ref>{{cite journal |last1=Uchiyama |first1=Mitsuru |title=The principal inverse of the gamma function |date=April 2012 |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 |doi-access=free }}</ref>

[[File:Inverse Gamma Function.png|thumb]]

[[File:Inverse gamma function in complex plane.png|Plot of inverse gamma function in the complex plane |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 |issue=2 |pages=251–267 |doi=10.1007/s00365-014-9239-1 |arxiv=1309.2167 |s2cid=253898042 |url=https://link.springer.com/article/10.1007/s00365-014-9239-1}}</ref>