Inverse 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 $\Gamma(y)=x$. 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 $$b \geqq 0$$.

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 ($$n\ge 0$$) $$ \Gamma_{n}^{-1}(z)=-n+\frac{\left(-1\right)^{n}}{n!z}+\frac{\psi^{(0)}\left(n+1\right)}{\left(n!z\right)^2}+\frac{\left(-1\right)^{n}\left(\pi^{2}+9\psi^{(0)}\left(n+1\right)^{2}-3\psi^{(1)}\left(n+1\right)\right)}{6\left(n!z\right)^3}+O\left(\frac{1}{z^{4}}\right)\,,$$ where $$\psi^{(n)}(x)$$ is the polygamma function.