Hyperasymptotics for integrals with saddles

Abstract

Integrals involving exp {-kf(z)}, where $|$k$|$ is a large parameter and the contour passes through a saddle of f(z), are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles, through which the contour does not pass. These contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent. The refinement is achieved by means of an exact `resurgence relation', expressing the original integral as its truncated saddle-point asymptotic expansion plus a remainder involving the integrals through certain `adjacent' saddles, determined by a topological rule. Iteration of the resurgence relation, and choice of truncation near the least term of the original series, leads to a representation of the integral as a sum of contributions associated with `multiple scattering paths among the saddles. No resummation of divergent series is involved. Each path gives a `hyperseries', depending on the terms in the asymptotic expansions for each saddle (these depend on the particular integral being studied and so are non-universal), and certain `hyperterminant' functions defined by integrals (these are always the same and hence universal). Successive hyperseries get shorter, so the scheme naturally halts. For two saddles, the ultimate error is approximately $\varepsilon ^{2.386}$, where $\varepsilon $ (proportional to exp (-A$|$k$|$) where A is a positive constant), is the error in optimal truncation of the original series. As a numerical example, an integral with three saddles is computed hyperasymptotically.