Asymptotic expansions of oscillatory integrals with complex phase

TL;DR

This paper extends the theory of asymptotic expansions for saddle point integrals with complex phases, generalizing classical Laplace and Fourier integral results using contour shifting in complex space.

Contribution

It introduces a method for asymptotic analysis of oscillatory integrals with complex phases, broadening the scope beyond real or purely imaginary phases.

Findings

Results analogous to Laplace and Fourier integrals for complex phases

Method based on contour shifting in complex space

Applications to asymptotic enumeration

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 enumeration.