Inverse gamma function: Difference between revisions

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

{{Orphan|date=July 2023}}

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([\Gamma(beta, +\alphainfty\right)=</math> 0.8856031...and image on the real interval <math>\left[\alpha, +\infty\right)</math>, where <math>\alphabeta =1.4616321.. 0.8856031\ldots</math><ref>{{oeis|A030171}}</ref> is the uniqueminimum positivevalue numberof suchthe thatgamma function on the positive real axis and <math>\psialpha = \Gamma^{-1}(\alphabeta) =0 1.4616321\ldots</math> is the ___location of that minimum.<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|doi-access=free }}</ref> (where <math>\psi(x)</math> is the [[digamma function]]).

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