# Inverse gamma function

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Inverse_gamma_function
> Markdown URL: https://mediated.wiki/source/Inverse_gamma_function.md
> Source: https://en.wikipedia.org/wiki/Inverse_gamma_function
> Source revision: 1354007385
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Inverse of the gamma function}}
{{Distinguish|Inverse-gamma distribution|Reciprocal gamma function}}
{{multiple image
 | total_width = 500
 | image1 = Inverse Gamma Function.png
 | caption1 = Graph of an inverse gamma function
 | image2 = Inverse gamma function in complex plane.png
 | caption2 = Plot of inverse gamma function in the complex plane
}}

In [mathematics](/source/mathematics), the '''inverse gamma function''' <math>\Gamma^{-1}(x)</math> is the [inverse function](/source/inverse_function) of the [gamma function](/source/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. |author-link1=Jonathan Borwein |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>

== 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 display="block">\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>\mu (t)</math> is a [Borel measure](/source/Borel_measure) such that <math display="block">\int_{-\infty}^{\Gamma\left(\alpha\right)}\left(\frac{1}{t^{2}+1}\right)d\mu(t)<\infty \,,</math> and <math>a</math> and <math>b</math> are real numbers with <math>b \geqq 0</math>.

== Approximation ==
To compute the branches of the inverse gamma function one can first compute the [Taylor series](/source/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 conference |first1=Robert M.|last1=Corless |first2=Folitse Komla|last2=Amenyou |last3=Jeffrey |first3=David  |book-title=2017 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC) |title=Properties and Computation of the Functional Inverse of Gamma |conference=International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC) |date=2017 |pages=65 |doi=10.1109/SYNASC.2017.00020|isbn=978-1-5386-2626-9 |s2cid=53287687 }}</ref>

<math display="block"> \Gamma^{-1}\left(x\right)\approx\alpha+\sqrt{\frac{2\left(x-\Gamma\left(\alpha\right)\right)}{\psi^{\left(1 \right)}\left(\alpha \right)\Gamma\left(\alpha\right)}}.</math> 

where <math> \psi^{\left(1 \right)} \left(x \right)</math> is the [trigamma function](/source/trigamma_function). The inverse gamma function also has the following [asymptotic formula](/source/asymptotic_formula)<ref>{{cite thesis |type=MS |last1=Amenyou |first1=Folitse Komla |last2=Jeffrey |first2=David |title="Properties and Computation of the inverse of the Gamma Function" |date=2018 |pages=28  |url=https://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=7340&context=etd}}</ref>
<math display="block"> \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>
where <math>W_0(x)</math> is the [Lambert W function](/source/Lambert_W_function). The formula is found by inverting the [Stirling approximation](/source/Stirling's_approximation), and so can also be expanded into an asymptotic series.

=== Series expansion ===
To obtain a series expansion of the inverse gamma function one can first compute the series expansion of the [reciprocal gamma function](/source/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 journal |last1=Couto |first1=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 display="block"> \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>
where <math>\psi^{(n)}(x)</math> is the [polygamma function](/source/polygamma_function).

== References ==
{{reflist}}

Category:Gamma and related functions

{{mathanalysis-stub}}

---
Adapted from the Wikipedia article [Inverse gamma function](https://en.wikipedia.org/wiki/Inverse_gamma_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Inverse_gamma_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.