[QUE/SM-01005] SM-PROBLEM

Using the identity $\Gamma(N\,+\,1)\,=\,N!=\,\int_0^\infty x^Ne^{-x} dx $
and write the integral in the form
$$ \int_0^\infty e^{Ng(x)}dx $$
Find the maximum of $g(x)$ ( which occurs say at $x_0$ ). Assume the integral
is dominated by the contribution from the neighbourhood of $x_0$ for large N. Expanding $g(x)$ up to second order in $(x-x_0)$ derive the Stirling's approximation ( for $n\rightarrow\,\infty$)
$$ \rm{ln}N!\,=\,N\rm{ln}N\,-\,N\,+\,\frac{1}{2}\rm{ln}(2\pi N)$$

( More exact formula , just for information, is
$$ N!\,=\,\left(\sqrt{2\pi}\right)e^{-N}N^{N+1/2}\left[1\,+\,\frac{1}{12N}\,+\,\frac{1}{288N^2}\,+\,\ldots\right]$$