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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 of f(z). Here we derive analogous results for integrals with integrands of a similar form, but whose local expansions in powers of 1/k are made about the finite endpoints 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 'adjacent' saddles. The saddles adjacent to the endpoint are determined by a topological rule. If the least term of the endpoint expansion is the N sub o (k)th, summing to here calculates the endpoint integral up to an error of approximately exp (-N0 (k)). We develop a scheme, involving iteration of the new resurgence relations with a similar one derived in BH, which can reduce this error down to exp(-2.386N0(k)). This 'hyperasymptotic' formalism parallels that of BH and incorporates automatically any change in the complete asymptotic expansion as the endpoint moves in the complex plane, provided that it does not coincide with other saddles. We illustrate the analytical and numerical use of endpoint hyperasymptotics by application to the complementary error function erfc(x) and a constructed 'incomplete' Airy function.
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