Inverse gamma function: Difference between revisions

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{{AFC comment|1=Users on the Gamma function page suggested that this should have its own page instead of being added as a new section on that page. See the talk page on [[Gamma function]] Also, the existing references have been improved and new ones have been added. [[User:Onlineuser577215|Onlineuser577215]] ([[User talk:Onlineuser577215|talk]]) 8:58, 3 May 2023 (UTC)}}

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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 |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=MITSURUMitsuru |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]]).

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==== Definition ====

The inverse gamma function may be defined by the following integral representation <ref>{{cite journal |last1=PedersePedersen |first1=Henrik |title="Inverses of gamma functions" |journal=Constructive Approximation |date=9 SepSeptember 2013 |pagesvolume=7 |doi=10.1007/s00365-014-9239-1 |url=https://link.springer.com/article/10.1007/s00365-014-9239-1}}</ref>
<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>

Where <math>\int_{-\infty}^{\Gamma\left(\alpha\right)}\left(\frac{1}{t^{2}+1}\right)d\mu(t)<\infty</math>, and a and b are real numbers with <math>b\geqq0</math>, and <math>\mu (t)</math> is the [[Borel measure|Borel Meausure]].

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\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]] <ref>{{Citecite journal |lastlast1=Amenyou |firstfirst1=Folitse Komla |last2=Jeffrey |first2=David |title="Properties and Computation of the Inverseinverse of the Gamma functionFunction" |date=2018 |pages=28|doi=10.1109/SYNASC.2017.00020 |url=https://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=7340&context=etd |journal=Western:Graduate & Postdoctoral Studies}}}</ref>

<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>

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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 nth''n'' th branch <math>\Gamma_{n}^{-1}(z)</math> of the inverse gamma function (<math>n\ge 0</math>) <ref>{{Cite web |last=Couto |first=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 |url-status=live |place=Section 8 |publication-place=Maple Conference Proceedings}}</ref>:

<math>\GammaGamma_{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>

Where <math>\psi^{(n)}(x)</math> is the [[polygamma function]].