Asymptotic expansions of oscillatory integrals with complex phase

Abstract

We consider saddle point integrals in d variables whose phase func- tions are 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 proofs are via contour shifting in complex d-space. This work is motivated by applications to asymptotic enumeration. arise in many areas of mathematics. There are many variations. This integral may involve one or more variables; the variables may be real or complex; the integral may be global or taken over a small neighborhood or oddly shaped set; varying degrees of smoothness may be assumed; and varying degrees of degeneracy may be allowed near the critical points of the phase function, φ. Often what is sought is a leading order estimate of I(λ) as the positive real parameter λ tends to ∞ ,o r an asymptotic series I(λ) ∼ � n cngn(λ), where {gn} is a sequence of elementary functions and the expansion is possibly nowhere convergent, but satisfies