Inverse gamma function

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 ${\displaystyle \psi (\alpha )=0}$ ^[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.

Series Expansion

To obtain a series expansion of the inverse gamma function one can first compute the series expansion of the reciprocal gamma function ${\displaystyle {\frac {1}{\Gamma (x)}}}$ near the poles at the negative integers, and then invert the series.

Setting ${\displaystyle z={\frac {1}{x}}}$ then yields, for the n th branch ${\displaystyle \Gamma _{n}^{-1}(z)}$ of the inverse gamma function (${\displaystyle n\geq 0}$) ^[6]

${\displaystyle \Gamma _{n}^{-1}(z)=-n+{\frac {\left(-1\right)^{n}}{n!z}}+{\frac {\psi ^{(0)}\left(n+1\right)}{\left(n!z\right)^{2}}}+{\frac {\left(-1\right)^{n}\left(\pi ^{2}+9\psi ^{(0)}\left(n+1\right)^{2}-3\psi ^{(1)}\left(n+1\right)\right)}{6\left(n!z\right)^{3}}}+O\left({\frac {1}{z^{4}}}\right)}$

Where ${\displaystyle \psi ^{(n)}(x)}$ is the polygamma function.

References

1. Borwein, Jonathan M.; Corless, Robert M. (2017). "Gamma and Factorial in the Monthly". The American Mathematical Monthly 125.5. arXiv:1703.05349.
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. Pedersen, Henrik (9 September 2013). ""Inverses of gamma functions"". Constructive Approximation. 7 (2): 251–267. arXiv:1309.2167. doi:10.1007/s00365-014-9239-1. S2CID 253898042.
4. Corless, Robert M.; Amenyou, Folitse Komla; Jeffrey, David (2017). "Properties and Computation of the Functional Inverse of Gamma". 2017 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). p. 65. doi:10.1109/SYNASC.2017.00020. ISBN 978-1-5386-2626-9. S2CID 53287687. {{cite book}}: |journal= ignored (help)
5. Amenyou, Folitse Komla; Jeffrey, David (2018). "Properties and Computation of the inverse of the Gamma Function" (MS). p. 28.
6. Couto, Ana Carolina Camargos; Jeffrey, David; Corless, Robert (November 2020). "The Inverse Gamma Function and its Numerical Evaluation". Maple Conference Proceedings. Section 8.