Inverse gamma function: Difference between revisions

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{{Short description|Inverse of the gamma function}}

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

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== 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]] 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]] 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)}{\Psipsi^{\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]]. The inverse gamma function also has the following [[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]]. The formula is found by inverting the [[Stirling's approximation|Stirling approximation]], and so can also be expanded into an asymptotic series.

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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]].

== References ==

{{reflist}}

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[[Category:Gamma and related functions]]

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