Robin Pemantle and I have just submitted this paper, which fills in some gaps from our previous work. There is still more to be done along these lines – asymptotics of integrals whose stationary points are not isolated or are at a vertex of a simplex, for example, requires more work.
Abstract: We consider saddle point integrals in d variables whose phase function is
neither real nor purely imaginary. Results analogous to those for Laplace (real
phase) and Fourier (imaginary phase) integrals hold whenever the phase function
is analytic and nondegenerate. These results generalize what is well known for
integrals of Laplace and Fourier type. The method is via contour shifting in
complex d-space. This work is motivated by applications to asymptotic