This is an analysis question I remember thinking about in high school. Reading some of the other topics here reminded me of this, and I'd like to hear other people's solutions to this.
We have the gamma function, which has a fairly elementary form as we all know,
$\Gamma(z) = \int_0^\infty e^{-t} t^{z-1} dt = \int_0^1 \left[ \ln(t^{-1}) \right]^{z-1}$
Which satisfies of course, $\Gamma(n) = (n-1)!$, $n\in \mathbb{N}$, and the various recurrence relations and other identities that we can all look up on wikipedia or mathwolrd or wherever. We note that the gamma function is increasing on the interval $[a,\infty]$ where $a\approx 1.46163$.
The question is–can we come up with an explicit inverse function to the gamma function on this interval which looks similarly simple?
My techniques at the time were to write down a differential equation that the inverse would satisfy, and solve it, which I could do in terms of a power series expansion (being in high school, ignoring the issues of convergence) to get an approximate solution. But I was never able to get a very nice looking or exact solution. I have a few more sophisticated tricks now to do this, but I would be interested to see how people with more experience with these kinds of questions would go about answering this.
The gamma function also satisfies a reasonable number of somewhat interesting looking functional relations like $\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z)$. Does the inverse function satisfy any similar relations?
Best Answer
David Cantrell gives a good approximation of $\Gamma^{-1}(n)$ on this page.
I'll copy the result here in case that page ever goes down:
$k$ = the positive zero of the digamma function, approximately $1.461632$
$c$ = $\sqrt{2\pi}/e - \Gamma(k)$, approximately $0.036534$
$L(x)$ = $\ln(\frac{x+c}{\sqrt{2\pi}})$
$W(x)$ = Lambert W function
$ApproxInvGamma(x)$ = $L(x)/W(\frac{L(x)}{e}) + \frac{1}{2}$