Hyperasymptotics for multidimensional integrals, exact remainder terms and the global connection problem

The method of steepest descents for single dimensional Laplace-type integrals involving an asymptotic parameter k was extended by Berry & Howls in 1991 to provide exact remainder terms for truncated asymptotic expansions in terms of contributions from certain non-local saddlepoints. This led to an improved asymptotic expansion (hyperasymptotics) which gave exponentially accurate numerical and analytic results, based on the topography of the saddle distribution in the single complex plane of the integrand. In this paper we generalize these results to similar well-behaved multidimensional integrands with quadratic critical points, integrated over infinite complex domains. As previously pointed out the extra complex dimensions give rise to interesting problems and phenomena. First, the conventionally defined surfaces of steepest descent are no longer unique. Second, the Stokes's phenomenon (whereby contributions from subdominant saddles enter the asymptotic representation) is of codimension one. Third, we can collapse the representation of the integral onto a single complex plane with branch cuts at the images of critical points. The new results here demonstrate that dimensionality only trivially affects the form of the exact multidimensional remainder. Thus the growth of the late terms in the expansion can be identified, and a hyperasymptotic scheme implemented. We show by a purely algebraic method how to determine which critical points contribute to the remainder and hence resolve the global connection problem, Riemann sheet structure and homology associated with the multidimensional topography of the integrand.