asymptotics - Growth of $Gamma(n,n)$

How can you get the asymptotics for the growth of $\Gamma(n,n)$?

$$\Gamma(n,n) = \int_n^\infty x^{n-1} \exp(-x) \mathrm{d}x$$

Answer

You can get it by a simple rescaling of the integral. Let $x=n t$; then

$$\begin{align}\Gamma(n,n) &= n^n \int_1^{\infty} dt \: t^{n-1} e^{-n t}\\ &= n^n \int_1^{\infty} \frac{dt}{t} e^{n (\log{t}-t)} \\ &= n^n e^{-n} \int_0^{\infty} \frac{du}{1+u} e^{n [\log{(1+u)}-u]} \end{align}$$

The contribution to this integral is dominated by that near $u=0$ as $n \rightarrow \infty$. We may then Taylor expand the term in the exponential and see that the leading asymptotic behavior of the integral is

$$\Gamma(n,n) \sim n^n e^{-n} \int_0^{\infty} du \: e^{-n u^2/2} = n^n e^{-n} \sqrt{\frac{\pi}{2 n}}$$

We may find further terms in the asymptotic behavior by Taylor expanding the higher-order terms in the exponential and the term outside the exponential. The next higher order term is

$$\left ( 1 + \frac{n u^3}{3} \right ) (1-u) \sim 1-u +\frac{n u^3}{3}$$

Evaluating the integrals that result from this expansion, we find the next higher term in the expansion:

$$\Gamma(n,n) = n^n e^{-n} \left [ \sqrt{\frac{\pi}{2 n}} - \frac{1}{3 n} + O\left ( n^{-3/2} \right ) \right ]$$