Inverse gamma function: Difference between revisions

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

{{Orphan|date=July 2023}}

In [[mathematics]], the '''inverse gamma function''' <math>\Gamma^{-1}(x)</math> is the [[inverse function]] of the [[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>{{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 |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>

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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|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 ==

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 bookconference |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) |chaptertitle=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>