Inverse gamma function

• Comment: Users on the Gamma function page suggested that this should have its own page instead of being added as a new section on that page. See the talk page on Gamma function. Onlineuser577215 (talk) 8:58, 3 May 2023 (UTC)

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Not to be confused with Inverse-gamma distribution or Reciprocal gamma function.

In mathematics, the inverse gamma function ${\displaystyle \Gamma ^{-1}(x)}$ is the inverse function of the gamma function. In other words, it is the function satisfying ${\textstyle \Gamma (y)=x}$. For example, ${\displaystyle \Gamma ^{-1}(24)=5}$ ^[1]. Usually, the inverse gamma function refers to the principal branch on the interval ${\displaystyle \left(\Gamma (\alpha )=0.8856031...,\infty \right)}$ where ${\displaystyle \alpha =1.4616321...}$ is the unique positive number such that ${\displaystyle \Psi (x)=0}$ ^[2] where ${\displaystyle \Psi (x)}$ is the digamma function.

Definition

The inverse gamma function may be defined by the following integral representation^[3]${\displaystyle \Gamma ^{-1}(x)=a+bx+\int _{-\infty }^{\Gamma (\alpha )}\left({\frac {1}{x-t}}-{\frac {t}{t^{2}-1}}\right)d\mu (t)}$

Where ${\displaystyle \int _{-\infty }^{\Gamma \left(\alpha \right)}\left({\frac {1}{t^{2}+1}}\right)d\mu (t)<\infty }$, and a and b are real numbers with ${\displaystyle b\geqq 0}$, and ${\displaystyle \mu (t)}$ is the Borel Meausure.

Approximation

To compute the branches of the inverse gamma function one can first compute the Taylor series of ${\displaystyle \Gamma (x)}$ near ${\displaystyle \alpha }$. The series can then be truncated and inverted, which yields successively better approximations to ${\displaystyle \Gamma ^{-1}(x)}$. For instance, we have the quadratic approximation^[4]

${\displaystyle \Gamma ^{-1}\left(x\right)\approx \alpha +{\sqrt {\frac {2\left(x-\Gamma \left(\alpha \right)\right)}{\Psi \left(1,\ \alpha \right)\Gamma \left(\alpha \right)}}}.}$

The inverse gamma function also has the following asymptotic formula^[5]

${\displaystyle \Gamma ^{-1}(x)\sim {\frac {1}{2}}+{\frac {\ln \left({\frac {x}{\sqrt {2\pi }}}\right)}{W_{0}\left(e^{-1}\ln \left({\frac {x}{\sqrt {2\pi }}}\right)\right)}}}$

Where ${\displaystyle W_{0}(x)}$ is the Lambert W function. The formula is found by inverting the Stirling approximation, and so can also be expanded into an asymptotic series.

References

1. Borwein, Corless (2017). "Gamma and Factorial in the Monthly". arXiv:1703.05349. {{cite journal}}: Cite journal requires |journal= (help)
2. Uchiyama, MITSURU (April 2012). "The principal inverse of the gamma function". Proceedings of the American Mathematical Society. 140 (4): 1347. doi:10.1090/S0002-9939-2011-11023-2. JSTOR 41505586. S2CID 85549521. Retrieved 20 March 2023.
3. Pederse, Henrik (9 Sep 2013). "Inverses of gamma functions". Constructive Approximation: 7. doi:10.1007/s00365-014-9239-1.
4. Corless; Folitse; Jeffrey (2017). "Properties and Computation of the Functional Inverse of Gamma". SYNASC: 65. doi:10.1109/SYNASC.2017.00020.
5. Amenyou, Komla. "Properties and Computation of the Inverse of the Gamma function". Western:Graduate & Postdoctoral Studies.