Inverse gamma function: Difference between revisions

Revision as of 15:46, 24 May 2023

Revision as of 11:55, 12 June 2023

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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 |lastlast1=Borwein |firstfirst1= 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 |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>

<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 <ref>{{cite journal |first1=Robert M.|last1=Corless |first2=Folitse Komla|last2=Amenyou |last3=Jeffrey |first3=David |title=Properties and Computation of the Functional Inverse of Gamma |journal=SYNASC |date=2017 |pages=65 |doi=10.1109/SYNASC.2017.00020|isbn=978-1-5386-2626-9 |s2cid=53287687 }}</ref>

<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 ''n'' th branch <math>\Gamma_{n}^{-1}(z)</math> of the inverse gamma function (<math>n\ge 0</math>) <ref>{{Cite webjournal |lastlast1=Couto |firstfirst1=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>