Inverse gamma function

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

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

Graph of an inverse gamma function

Plot of inverse gamma function in the complex plane

Definition

The inverse gamma function may be defined by the following integral representation $$\Gamma ^{-1}(x)=a+bx+\int _{-\infty }^{\Gamma (\alpha )}\left({\frac {1}{x-t}}-{\frac {t}{t^{2}-1}}\right)d\mu (t)\,,$$ where $\mu (t)$ is a Borel measure such that $$\int _{-\infty }^{\Gamma \left(\alpha \right)}\left({\frac {1}{t^{2}+1}}\right)d\mu (t)<\infty \,,$$ and $a$ and $b$ are real numbers with $b\geqq 0$.

Approximation

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

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

The inverse gamma function also has the following asymptotic formula $$\Gamma ^{-1}(x)\sim {\frac {1}{2}}+{\frac {\ln \left({\frac {x}{\sqrt {2\pi }}}\right)}{W_{0}\left(e^{-1}\ln \left({\frac {x}{\sqrt {2\pi }}}\right)\right)}}\,,$$ where $W_{0}(x)$ 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 ${\frac {1}{\Gamma (x)}}$ near the poles at the negative integers, and then invert the series.

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