Inverse gamma function: Difference between revisions

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

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

{{multiple image

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 |last1=Borwein |first1= 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(\alpha)=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

| total_width = 500

|jstor=41505586 |s2cid=85549521 |access-date=20 March 2023}}</ref> (where <math>\psi(x)</math> is the [[digamma function]]).

[[File: | image1 = Inverse Gamma Function.png|thumb]]

| 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]], the '''inverse gamma function''' <math>\Gamma^{-1}(x)</math> is the [[inverse function]] of the [[gamma function]]. In other words, it<math>y is= the function\Gamma^{-1}(x)</math> satisfyingwhenever <math display="inline">\Gamma(y)=x</math>. For example, <math>\Gamma^{-1}(24)=5</math> .<ref>{{Citecite 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 |yearpages=2017 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><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 |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

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

|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>

Wherewhere <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\geqq0</math>, and <math>\mugeqq (t)0</math> is the [[Borel measure|Borel Meausure]].

==== Approximation ====

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]].

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 journalconference |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 |journalconference=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>

==== 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>

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>

<math display="block"> \Gamma^{-1}\left(x\right)\approxsim\alphafrac{1}{2}+\sqrtfrac{\frac{2ln\left(\frac{x-}{\Gammasqrt{2\left(\alpha\right)pi}}\right)}{\PsiW_{0}\left(e^{-1,}\ ln\alphaleft(\right)frac{x}{\Gammasqrt{2\left(pi}}\alpharight)\right)}}.\,,</math>

Wherewhere <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.

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>\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.

'''Series Expansion'''

=== Series expansion ===

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 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>

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

<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>

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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