Inverse gamma function

Short description: Inverse of the gamma function

Graph of an inverse gamma function

Plot of inverse gamma function in the complex plane

In mathematics, the inverse gamma function ${\mathrm{\Gamma }}^{-1}(x)$ is the inverse function of the gamma function. In other words, $y={\mathrm{\Gamma }}^{-1}(x)$ whenever $\mathrm{\Gamma }(y)=x$. For example, ${\mathrm{\Gamma }}^{-1}(24)=5$. Usually, the inverse gamma function refers to the principal branch with domain on the real interval $[\beta ,+\mathrm{\infty })$ and image on the real interval $[\alpha ,+\mathrm{\infty })$, where $\beta =0.8856031\dots$ is the minimum value of the gamma function on the positive real axis and $\alpha ={\mathrm{\Gamma }}^{-1}(\beta )=1.4616321\dots$ is the location of that minimum.

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

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 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 n th branch ${\mathrm{\Gamma }}_n^{-1}(z)$ of the inverse gamma function ($n\ge 0$) $${\mathrm{\Gamma }}_n^{-1}(z)=-n+\frac{{(-1)}^n}{n!z}+\frac{{\psi }^{(0)}(n+1)}{{(n!z)}^2}+\frac{{(-1)}^n({\pi }^2+9{\psi }^{(0)}{(n+1)}^2-3{\psi }^{(1)}(n+1))}{6{(n!z)}^3}+O\left(\frac{1}{z^4}\right),$$ where ${\psi }^{(n)}(x)$ is the polygamma function.