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Globally Exact Asymptotics for Integrals with Arbitrary Order Saddles

Bennett, T., Howls, C.J., Nemes, G., Olde Daalhuis, A.B. (2018) Globally Exact Asymptotics for Integrals with Arbitrary Order Saddles. SIAM Journal on Mathematical Analysis, 50 (2). pp. 2144-2177. ISSN 0036-1410. (doi:10.1137/17M1154217) (KAR id:68713)

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https://dx.doi.org/10.1137/17M1154217

Abstract

We derive the first exact, rigorous but practical, globally valid remainder terms for asymptotic expansions about saddles and contour endpoints of arbitrary order degeneracy derived from the method of steepest descents. The exact remainder terms lead naturally to sharper novel asymptotic bounds for truncated expansions that are a significant improvement over the previous best existing bounds for quadratic saddles derived two decades ago. We also develop a comprehensive hyperasymptotic theory, whereby the remainder terms are iteratively reexpanded about adjacent saddle points to achieve better-than-exponential accuracy. By necessity of the degeneracy, the form of the hyperasymptotic expansions is more complicated than in the case of quadratic endpoints and saddles and requires generalizations of the hyperterminants derived in those cases. However, we provide efficient methods to evaluate them, and we remove all possible ambiguities in their definition. We illustrate this approach for three different examples, providing all the necessary information for the practical implementation of the method.

Item Type: Article
DOI/Identification number: 10.1137/17M1154217
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Tom Bennett
Date Deposited: 20 Aug 2018 08:52 UTC
Last Modified: 14 Jan 2020 09:51 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/68713 (The current URI for this page, for reference purposes)
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