General Question: Is there a reference where the saddle point approximation is applied to multiple contour integrals?
In particular, say we have the integral $$ I_N = \frac{1}{(2\pi i)^N} \oint \left[\prod_{\ell=1}^N \frac{dq_{\ell}}{q_{\ell}^2}\right] \left(\sum_{j=1}^{N} q_{j}\right)^{N},$$ where we are applying $N$ contour integrations in sequence on curves that circle the origin in the complex plane. Using the multinomial theorem one can show that $I_N = N!$.
Using elementary symmetric polynomials, one can also show that $I_N$ can be written as
\begin{equation} I_N = \frac{1}{(2\pi i)^{2N}} \oint \left[\prod_{\ell=1}^N\frac{dz_{\ell}}{z_{\ell}^2} \right] \left[\prod_{m=1}^N\frac{dq_{m}}{q_{m}^2} \right] \prod_{i, j=1}^N\Big(1 + z_i q_j\Big), \label{eq:INcont} \end{equation}
where we are now applying $2N$ contour integrations.
Specific Question: Is it possible to apply a multi-dimensional saddle point approximation to either form of $I_N$ to derive some form of Stirling's approximation (i.e., $N!\simeq \sqrt{2\pi N} (N/e)^N$)?
(Reposted from stackexchange due to no answers.)
