It is well known that the solid angle in an euclidian space of $d$ dimensions ($d = 2 n$ or $d = 2 n + 1$, where $n = 1, 2, 3, \dots, \infty$) is given by these formulae: \begin{align}\tag{1} \Omega_{2 n} &= \frac{1}{(n - 1)!} \, 2 \pi^n, \qquad &\Omega_{2 n + 1} &= \frac{2^{2 n} \, n!}{(2 n)!} \, 2 \pi^n. \end{align} For examples: $\Omega_1 = 2$, $\Omega_2 = 2 \pi$, $\Omega_3 = 4 \pi$, $\Omega_4 = 2 \pi^2$. Oddly, these formulae have a maximum value for $d = 7$: $$\tag{2} \Omega_7 = \frac{16 \pi^3}{15}, $$ and $\Omega_d \rightarrow 0$ when $d \rightarrow \infty$. We could also use the Gamma function $\Gamma(n)$ to get a smooth version of (1), instead of a discrete one, and plot some nice graphics with Mathematica: $$\tag{3} \Omega(d) = \frac{2 \sqrt{\pi}^d}{\Gamma(d/2)}. $$ My question is simple:
What is the simplest asymptotic version of (1) or (3), for very large values of $d$?
I tried using the Stirling formula into (3) but I'm not getting a good result.
Hmm, the Asymptotic function of Mathematica gave me this expression for large $d$: $$ \Omega(d) \approx \frac{1}{6} \, \frac{6 d - 1}{\sqrt{\pi d}} \, \sqrt{\frac{2 \pi e}{d}}^{\, d} \approx \sqrt{\frac{d}{\pi}} \, \sqrt{\frac{2 \pi e}{d}}^{\, d}. $$ This appears to be what I'm looking for.