The integral $I(\lambda) = \int_{\mathbb{R}^d} \mathrm{d}x\, g(x)\, e^{-\lambda f(x)}$ can be approximated for $\lambda \rightarrow \infty$ using Laplace's method if $f(x)$ has a nondegenerate critical point $x_0$. That is, the determinant of the Hessian matrix of $f$ at $x_0$ does not vanish. Here I am assuming that both $g$ and $f$ can be expressed using a Taylor series.
The question is, how can one obtain an asymptotic approximation to the integral when $f$ has a single critical point, say the origin, at which the above determinant vanishes? In particular, I have $f(x) = \exp(-N |\sum_j (c_j + ic_{j+1})(x_j + ix_{j+1})|^2)$.
I know there is a book by Paris and Kaminski, in which Mellin-Barnes integrals and the Newton polygon are used for obtaining asymptotic approximations to this kind of integrals. Hopefully someone has experience with this and can provide some practical guidance on how to approximate the above integral.