Hyperasymptotics for Integrals with Finite Endpoints

C. J. Howls
1992 Proceedings of the Royal Society A  
Physics L a b o r a t o r y , Tyndall , Bristol BS8 , U.K. Berry & Howls (1991) (hereinafter called BH) refined the method of steepest descent to study exponentially accurate asymptotics of a general class of integrals involving exp{-kf(z)} along doubly infinite contours in the complex plane passing over saddlepoints off(z). Here we derive analogous results for integrals with integrands of a similar form, but whose local expansions in powers of 1 are made about the finite endpoints of
more » ... points of semi-infinite contours of integration. We treat endpoints where f(z) behaves locally linearly or quadratically. Generically, local endpoint expansions made by the method of steepest descent diverge because of the presence of saddles of f(z). We derive 'resurgence relations' which express the original integral exactly as a truncated endpoint expansion plus a remainder, involving the global saddle structure of f(z) via integrals through certain 'ad jacen t' saddles. The saddles adjacent to the endpoint are determined by a topological rule. If the least term of the endpoint expansion is the iV0(A;)th, summing to here calculates the endpoint integral up to an error of approximately exp ( -N0( J c ) ) . We develo iteration of the new resurgence relations with a similar one derived in BH, which can reduce this error down to exp ( -2.386iV0(&)). This 'hyperasym ptotic' formalism parallels th a t of BH and incorporates autom atically any change in the complete asym ptotic expansion as the endpoint moves in the complex plane, provided th a t it does not coincide with other saddles. We illustrate the analytical and numerical use of endpoint hyperasym ptotics by application to the complementary error function erfc(x) and a constructed 'incom plete' Airy function.
doi:10.1098/rspa.1992.0156 fatcat:xcstzv73hbhnhiwi3nyjhxzi2y