Inverse gamma function

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

Definition

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

Series Expansions

Let ${\mathrm{\Gamma }}_n^{-1}(z)$ be the $n$-th branch of the gamma function, with ${\mathrm{\Gamma }}_0^{-1}(z)$ 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 $\frac{1}{\mathrm{\Gamma }(x)}$ near the zeros at the negative integers, and then invert the series.

Setting $z=\frac{1}{x}$ in the expansion then yields, for $n\ge 1$:$${\mathrm{\Gamma }}_n^{-1}(z)=-n+1-\frac{{(-1)}^n}{(n-1)!z}+\frac{{\psi }^{(0)}\left(n\right)}{(n-1){!}^2{z}^2}-\frac{{(-1)}^n({\pi }^2+9{\psi }^{(0)}{\left(n\right)}^2-3{\psi }^{(1)}\left(n\right))}{6(n-1){!}^3{z}^3}+O\left(\frac{1}{z^4}\right),$$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 Template:Mvar is analytic at a point Template:Mvar and $f^{\prime }(a)\ne 0$, then the inverse $w=g(z)$ is given by a power series

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

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

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

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

${\mathrm{\Gamma }}_1^{-1}(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{x^{-n}}{n!}{[\frac{d^{n-1}}{d{w}^{n-1}}\mathrm{\Gamma }(w+1{)}^n]}_{w=0}.$

Approximation

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

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

where ${\psi }^{\left(1\right)}\left(x\right)$ is the trigamma function. The inverse gamma function also has the following asymptotic formula $${\mathrm{\Gamma }}^{-1}(x)\sim \frac{1}{2}+\frac{\ln \left(\frac{x}{\sqrt{2\pi }}\right)}{W_0(e^{-1}\ln \left(\frac{x}{\sqrt{2\pi }}\right))},$$ 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.

References

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