Inverse Gamma function

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

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

Definition

The inverse gamma function may be defined by the following integral representation${\mathrm{\Gamma }}^{-1}(x)=a+bx+\underset{-\mathrm{\infty }}{\overset{\mathrm{\Gamma }(\alpha )}{\int }}(\frac{1}{x-t}-\frac{t}{t^2-1})d\mu (t)$

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

Approximation

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

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

The inverse gamma function also has the following asymptotic formula

${\mathrm{\Gamma }}^{-1}(x)\sim \frac{1}{2}+\frac{\ln \left(\frac{x}{\sqrt{2\pi }}\right)}{W_0(e^{-1}\ln \left(\frac{x}{\sqrt{2\pi }}\right))}$

Where $W_0(x)$ is the Lambert W function. The formula is found by inverting the Stirling's 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 $\frac{1}{\mathrm{\Gamma }(x)}$ near the poles at the negative integers, and then invert the series.

Setting $z=\frac{1}{x}$ then yields, for the nth branch of the inverse gamma function ($n\ge 0$) :

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

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

References

This article "Inverse Gamma function" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Inverse Gamma function. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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