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, ${\displaystyle y=\Gamma ^{-1}(x)}$ whenever ${\textstyle \Gamma (y)=x}$. For example, ${\displaystyle \Gamma ^{-1}(24)=5}$.^[1] Usually, the inverse gamma function refers to the principal branch with domain on the real interval ${\displaystyle \left[\beta ,+\infty \right)}$ and image on the real interval ${\displaystyle \left[\alpha ,+\infty \right)}$, where ${\displaystyle \beta =0.8856031\ldots }$^[2] is the minimum value of the gamma function on the positive real axis and ${\displaystyle \alpha =\Gamma ^{-1}(\beta )=1.4616321\ldots }$^[3] is the location of that minimum.^[4]

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^[5] $${\displaystyle \Gamma ^{-1}(x)=a+bx+\int _{-\infty }^{\Gamma (\alpha )}\left({\frac {1}{x-t}}-{\frac {t}{t^{2}-1}}\right)d\mu (t)\,,}$$ where ${\displaystyle \mu (t)}$ is a Borel measure such that $${\displaystyle \int _{-\infty }^{\Gamma \left(\alpha \right)}\left({\frac {1}{t^{2}+1}}\right)d\mu (t)<\infty \,,}$$ and ${\displaystyle a}$ and ${\displaystyle b}$ are real numbers with ${\displaystyle b\geqq 0}$.

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:^[6]

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

where ${\displaystyle \psi ^{\left(1\right)}\left(x\right)}$ is the trigamma function. The inverse gamma function also has the following asymptotic formula^[7] $${\displaystyle \Gamma ^{-1}(x)\sim {\frac {1}{2}}+{\frac {\ln \left({\frac {x}{\sqrt {2\pi }}}\right)}{W_{0}\left(e^{-1}\ln \left({\frac {x}{\sqrt {2\pi }}}\right)\right)}}\,,}$$ 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 (${\displaystyle n\geq 0}$)^[8] $${\displaystyle \Gamma _{n}^{-1}(z)=-n+{\frac {\left(-1\right)^{n}}{n!z}}+{\frac {\psi ^{(0)}\left(n+1\right)}{\left(n!z\right)^{2}}}+{\frac {\left(-1\right)^{n}\left(\pi ^{2}+9\psi ^{(0)}\left(n+1\right)^{2}-3\psi ^{(1)}\left(n+1\right)\right)}{6\left(n!z\right)^{3}}}+O\left({\frac {1}{z^{4}}}\right)\,,}$$ where ${\displaystyle \psi ^{(n)}(x)}$ is the polygamma function.