Inverse gamma function

Graph of an inverse gamma function

Plot of inverse gamma function in the complex plane

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]

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 ${\displaystyle b\geqq 0}$.

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

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

This can be rigorously justified by the Lagrange inversion theorem, which says that if ${\displaystyle z=f(w)}$ where f is analytic at a point a and ${\displaystyle f'(a)\neq 0}$, then the inverse ${\displaystyle w=g(z)}$ is given by a power series^[7]

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

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

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

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

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

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:^[8]

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

where ${\displaystyle \psi ^{\left(1\right)}\left(x\right)}$ is the trigamma function. The inverse gamma function also has the following asymptotic formula^[9] $${\displaystyle \Gamma ^{-1}(x)\sim {\frac {1}{2}}+{\frac {\ln \left({\frac {x}{\sqrt {2\pi }}}\right)}{W_{0}\left(e^{-1}\ln \left({\frac {x}{\sqrt {2\pi }}}\right)\right)}}\,,}$$ 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.

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