Globally exact asymptotics for integrals with arbitrary order saddles
Globally exact asymptotics for integrals with arbitrary order saddles
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.
integral asymptotics, saddle points, error bounds, hyperasymptotics, asymptotic expansions
2144-2177
Bennett, T.
17f74efb-6425-461a-9f4e-a445b4b0d25a
Howls, C.J.
66d3f0f0-376c-4f7a-a206-093935e6c560
Nemes, Gergo
0b417dcd-cbef-45c5-9778-54203d5b33e1
Olde Daalhuis, Adri
1dba32ab-7517-4606-961c-dfa805a72294
17 April 2018
Bennett, T.
17f74efb-6425-461a-9f4e-a445b4b0d25a
Howls, C.J.
66d3f0f0-376c-4f7a-a206-093935e6c560
Nemes, Gergo
0b417dcd-cbef-45c5-9778-54203d5b33e1
Olde Daalhuis, Adri
1dba32ab-7517-4606-961c-dfa805a72294
Bennett, T., Howls, C.J., Nemes, Gergo and Olde Daalhuis, Adri
(2018)
Globally exact asymptotics for integrals with arbitrary order saddles.
SIAM Journal on Mathematical Analysis, 50 (2), .
(doi: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.
Text
Globally Exact Asymptotics For Integrals With Arbitrary Order Saddles
- Accepted Manuscript
More information
Submitted date: 27 October 2017
Accepted/In Press date: 7 February 2018
e-pub ahead of print date: 17 April 2018
Published date: 17 April 2018
Additional Information:
Funding Information:
The first author was sponsored by an EPSRC studentship. The third and fourth authors were supported by a research grant (GRANT11863412/70NANB15H221) from the National Institute of Standards and Technology.
Funding Information:
∗Received by the editors October 27, 2017; accepted for publication (in revised form) February 7, 2018; published electronically April 17, 2018. http://www.siam.org/journals/sima/50-2/M115421.html Funding: The first author was sponsored by an EPSRC studentship. The third and fourth authors were supported by a research grant (GRANT11863412/ 70NANB15H221) from the National Institute of Standards and Technology. †SMSAS, University of Kent, Sibson Building, Parkwood Road, Canterbury CT2 7FS, UK (T.B.Bennett@kent.ac.uk). ‡Mathematical Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, UK (C.J.Howls@maths.soton.ac.uk). §School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Edinburgh EH9 3FD, UK (Gergo.Nemes@ed.ac.uk, A.OldeDaalhuis@ed.ac.uk).
Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.
Keywords:
integral asymptotics, saddle points, error bounds, hyperasymptotics, asymptotic expansions
Identifiers
Local EPrints ID: 418338
URI: http://eprints.soton.ac.uk/id/eprint/418338
ISSN: 1095-7154
PURE UUID: 33fabced-54e0-4556-8095-8f527d28d1d1
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Date deposited: 28 Feb 2018 17:32
Last modified: 16 Mar 2024 05:24
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Contributors
Author:
T. Bennett
Author:
Gergo Nemes
Author:
Adri Olde Daalhuis
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