Inverse gamma function: Difference between revisions

Revision as of 18:22, 27 April 2023

Revision as of 19:22, 27 April 2023

Line 1:

{{AFC submission|d|reason|Please discuss at the talk page for [[Gamma function]] whether this should be a section on that page or split off.|u=Onlineuser577215|ns=118|decliner=AngusWOOF|declinets=20230427192227|ts=20230427182236}} <!-- Do not remove this line! -->

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

{{AFC submission|d|v|u=Onlineuser577215|ns=118|decliner=Majash2020|declinets=20230412023636|small=yes|ts=20221206210833}} <!-- Do not remove this line! -->

{{Draft topics|mathematics}}

{{AfC topic|stem}}

{{AfC submission|||ts=20230427182236|u=Onlineuser577215|ns=118}}

{{AFC submission|d|v|u=Onlineuser577215|ns=118|decliner=Majash2020|declinets=20230412023636|ts=20221206210833}} <!-- Do not remove this line! -->

{{AFC comment|1=Please fix the references to this. The article is definitely notable, mostly the references need to be fixed [[User:Majash2020|Majash2020]] ([[User talk:Majash2020|talk]]) 02:36, 12 April 2023 (UTC)}}

Line 11 ⟶ 8:

----

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

{{Draft topics|mathematics}}

{{AfC topic|stem}}

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 |last=Borwein, Corless |title=Gamma and Factorial in the Monthly |journal=|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 the [[digamma function]] <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

Line 17 ⟶ 16:

==== Definition ====

The inverse gamma function may be defined by the following integral representation <ref>{{cite journal |last1=Pederse |first1=Henrik |title=Inverses of gamma functions |journal=Constructive Approximation |date=9 Sep 2013 |pages=7 |doi=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>

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]].

Line 24 ⟶ 23:

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 |last1=Corless |last2=Folitse |last3=Jeffrey |title=Properties and Computation of the Functional Inverse of Gamma |journal=SYNASC |date=2017 |pages=65 |doi=10.1109/SYNASC.2017.00020}}</ref>

<math>

\Gamma^{-1}\left(x\right)\approx\alpha+\sqrt{\frac{2\left(x-\Gamma\left(\alpha\right)\right)}{\Psi\left(1,\ \alpha\right)\Gamma\left(\alpha\right)}}.</math>