Hyperasymptotics
Hyperasymptotics
We develop a technique for systematically reducing the exponentially small (‘superasymptotic’) remainder of an asymptotic expansion truncated near its least term, for solutions of ordinary differential equations of Schrödinger type where one transition point dominates. This is achieved by repeatedly applying Borel summation to a resurgence formula discovered by Dingle, relating the late to the early terms of the original expansion. The improvements form a nested sequence of asymptotic series truncated at their least terms. Each such ‘hyperseries’ involves the terms of the original asymptotic series for the particular function being approximated, together with terminating integrals that are universal in form, and is half the length of its predecessor. The hyperasymptotic sequence is therefore finite, and leads to an ultimate approximation whose error is less than the square of the original superasymptotic remainder. The Stokes phenomenon is automatically and exactly incorporated into the scheme. Numerical computations confirm the efficacy of the technique.
Hyperasymptotcs, asymptotics, Exponential asymptotics, resurgence, resurgent function, superasymptotics, optimal truncation, exponential accuracy, numerical approximation, Stokes phenmenon, terminant, Dingle, Hyperterminant
653-668
Berry, Michael
ec39b1ad-7f54-4abf-9fcf-e5a3d1c2ab84
Howls, Christopher
66d3f0f0-376c-4f7a-a206-093935e6c560
8 September 1990
Berry, Michael
ec39b1ad-7f54-4abf-9fcf-e5a3d1c2ab84
Howls, Christopher
66d3f0f0-376c-4f7a-a206-093935e6c560
Berry, Michael and Howls, Christopher
(1990)
Hyperasymptotics.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 430 (1880), .
(doi:10.1098/rspa.1990.0111).
Abstract
We develop a technique for systematically reducing the exponentially small (‘superasymptotic’) remainder of an asymptotic expansion truncated near its least term, for solutions of ordinary differential equations of Schrödinger type where one transition point dominates. This is achieved by repeatedly applying Borel summation to a resurgence formula discovered by Dingle, relating the late to the early terms of the original expansion. The improvements form a nested sequence of asymptotic series truncated at their least terms. Each such ‘hyperseries’ involves the terms of the original asymptotic series for the particular function being approximated, together with terminating integrals that are universal in form, and is half the length of its predecessor. The hyperasymptotic sequence is therefore finite, and leads to an ultimate approximation whose error is less than the square of the original superasymptotic remainder. The Stokes phenomenon is automatically and exactly incorporated into the scheme. Numerical computations confirm the efficacy of the technique.
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Accepted/In Press date: 6 June 1990
Published date: 8 September 1990
Keywords:
Hyperasymptotcs, asymptotics, Exponential asymptotics, resurgence, resurgent function, superasymptotics, optimal truncation, exponential accuracy, numerical approximation, Stokes phenmenon, terminant, Dingle, Hyperterminant
Identifiers
Local EPrints ID: 496796
URI: http://eprints.soton.ac.uk/id/eprint/496796
ISSN: 1364-5021
PURE UUID: ae940b93-b524-4565-89d2-fdf0c3d71576
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Date deposited: 08 Jan 2025 07:07
Last modified: 10 Jan 2025 02:39
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Author:
Michael Berry
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