// Workers AI · dad joke modeWhat did inverse gamma function say? "I'm inverse-ly excited

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

Graph of an inverse gamma function

Plot of inverse gamma function in the complex plane

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.

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 $b\geqq 0$.

Series Expansions

Let $\Gamma _{n}^{-1}(z)$ be the $n$-th branch of the gamma function, with $\Gamma _{0}^{-1}(z)$ denoting the principal branch. 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 zeros at the negative integers, and then invert the series.

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

This can be rigorously justified by the Lagrange inversion theorem, which says that if $z=f(w)$ where f is analytic at a point a and $f'(a)\neq 0$, then the inverse $w=g(z)$ is given by a power series

$g(z)=a+\sum _{n=1}^{\infty }g_{n}{\frac {(z-f(a))^{n}}{n!}},\quad g_{n}=\lim _{w\to a}{\frac {d^{n-1}}{dw^{n-1}}}\left[\left({\frac {w-a}{f(w)-f(a)}}\right)^{n}\right].$

For example, for the first non principal branch, set $f(x)={\frac {1}{\Gamma (x)}}$ and $a=0$ such that $f(a)=0$. The formula gives

$g(z)=\sum _{n=1}^{\infty }{\frac {z^{n}}{n!}}\left[\lim _{w\to 0}{\frac {d^{n-1}}{dw^{n-1}}}\Gamma (w+1)^{n}\right].$

Now, assuming ${\frac {1}{\Gamma (x)}}$ is invertible when restricted to an appropriate region, setting $z={\frac {1}{x}}$ gives

$\Gamma _{1}^{-1}(x)=\sum _{n=1}^{\infty }{\frac {x^{-n}}{n!}}\left[{\frac {d^{n-1}}{dw^{n-1}}}\Gamma (w+1)^{n}\right]_{w=0}.$

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:

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

where $\psi ^{\left(1\right)}\left(x\right)$ is the trigamma function. 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.