Stirling's formula derived simply

Stirling’s formula, the asymptotic expansion of $n!$ for $n$ large, or of $\mathrm{\Gamma }(z)$ for $z\to \mathrm{\infty }$, is derived directly from the recursion equation $\mathrm{\Gamma }(z+1)=z\mathrm{\Gamma }(s)$ and the normalization condition $\Gamma ({1/2}) =\sqrt{\pi}$.