The swallowtail integral in the highly oscillatory region III
We consider the swallowtail integral Ψ(x,y,z):=∫∞−∞ei(t5+xt3+yt2+zt)dt for large values of |z| and bounded values of |x| and |y|. The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Universidad Pública de Navarra |
| Repositorio: | Academica-e. Repositorio Institucional de la Universidad Pública de Navarra |
| OAI Identifier: | oai:academica-e.unavarra.es:2454/40436 |
| Acceso en línea: | https://hdl.handle.net/2454/40436 |
| Access Level: | acceso abierto |
| Palabra clave: | Swallowtail integral Asymptotic expansions Modified saddle point method |
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The swallowtail integral in the highly oscillatory region IIIFerreira González, CheloLópez García, José LuisPérez Sinusía, EsterSwallowtail integralAsymptotic expansionsModified saddle point methodWe consider the swallowtail integral Ψ(x,y,z):=∫∞−∞ei(t5+xt3+yt2+zt)dt for large values of |z| and bounded values of |x| and |y|. The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the modified saddle point method introduced in López et al., A systematization of the saddle point method application to the Airy and Hankel functions. J Math Anal Appl. 2009;354:347–359. The analysis is more straightforward with this method and it is possible to derive complete asymptotic expansions of Ψ(x,y,z) for large |z| and fixed x and y. The asymptotic analysis requires the study of three different regions for argz separated by three Stokes lines in the sector −π<argz≤π. The asymptotic approximation is a certain combination of two asymptotic series whose terms are elementary functions of x, y and z. They are given in terms of an asymptotic sequence of the order O(z−n/12) when |z|→∞, and it is multiplied by an exponential factor that behaves differently in the three mentioned sectors. The accuracy and the asymptotic character of the approximations are illustrated with some numerical experiments.This research was supported by the Ministerio de Economía y Competitividad (MTM2017-83490-P) and the Universidad Pública de Navarra.Taylor & FrancisEstatistika, Informatika eta MatematikaInstitute for Advanced Materials and Mathematics - INAMAT2Estadística, Informática y MatemáticasUniversidad Pública de Navarra / Nafarroako Unibertsitate Publikoa2021info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttps://hdl.handle.net/2454/40436reponame:Academica-e. Repositorio Institucional de la Universidad Pública de Navarrainstname:Universidad Pública de NavarraInglésinfo:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P© 2021 Informa UK Limited, trading as Taylor & Francis Groupinfo:eu-repo/semantics/openAccessoai:academica-e.unavarra.es:2454/404362026-06-17T12:41:47Z |
| dc.title.none.fl_str_mv |
The swallowtail integral in the highly oscillatory region III |
| title |
The swallowtail integral in the highly oscillatory region III |
| spellingShingle |
The swallowtail integral in the highly oscillatory region III Ferreira González, Chelo Swallowtail integral Asymptotic expansions Modified saddle point method |
| title_short |
The swallowtail integral in the highly oscillatory region III |
| title_full |
The swallowtail integral in the highly oscillatory region III |
| title_fullStr |
The swallowtail integral in the highly oscillatory region III |
| title_full_unstemmed |
The swallowtail integral in the highly oscillatory region III |
| title_sort |
The swallowtail integral in the highly oscillatory region III |
| dc.creator.none.fl_str_mv |
Ferreira González, Chelo López García, José Luis Pérez Sinusía, Ester |
| author |
Ferreira González, Chelo |
| author_facet |
Ferreira González, Chelo López García, José Luis Pérez Sinusía, Ester |
| author_role |
author |
| author2 |
López García, José Luis Pérez Sinusía, Ester |
| author2_role |
author author |
| dc.contributor.none.fl_str_mv |
Estatistika, Informatika eta Matematika Institute for Advanced Materials and Mathematics - INAMAT2 Estadística, Informática y Matemáticas Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa |
| dc.subject.none.fl_str_mv |
Swallowtail integral Asymptotic expansions Modified saddle point method |
| topic |
Swallowtail integral Asymptotic expansions Modified saddle point method |
| description |
We consider the swallowtail integral Ψ(x,y,z):=∫∞−∞ei(t5+xt3+yt2+zt)dt for large values of |z| and bounded values of |x| and |y|. The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the modified saddle point method introduced in López et al., A systematization of the saddle point method application to the Airy and Hankel functions. J Math Anal Appl. 2009;354:347–359. The analysis is more straightforward with this method and it is possible to derive complete asymptotic expansions of Ψ(x,y,z) for large |z| and fixed x and y. The asymptotic analysis requires the study of three different regions for argz separated by three Stokes lines in the sector −π<argz≤π. The asymptotic approximation is a certain combination of two asymptotic series whose terms are elementary functions of x, y and z. They are given in terms of an asymptotic sequence of the order O(z−n/12) when |z|→∞, and it is multiplied by an exponential factor that behaves differently in the three mentioned sectors. The accuracy and the asymptotic character of the approximations are illustrated with some numerical experiments. |
| publishDate |
2021 |
| dc.date.none.fl_str_mv |
2021 |
| dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/acceptedVersion |
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article |
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acceptedVersion |
| dc.identifier.none.fl_str_mv |
https://hdl.handle.net/2454/40436 |
| url |
https://hdl.handle.net/2454/40436 |
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Inglés |
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Inglés |
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info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P |
| dc.rights.none.fl_str_mv |
© 2021 Informa UK Limited, trading as Taylor & Francis Group info:eu-repo/semantics/openAccess |
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© 2021 Informa UK Limited, trading as Taylor & Francis Group |
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openAccess |
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application/pdf |
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Taylor & Francis |
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Taylor & Francis |
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reponame:Academica-e. Repositorio Institucional de la Universidad Pública de Navarra instname:Universidad Pública de Navarra |
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Universidad Pública de Navarra |
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Academica-e. Repositorio Institucional de la Universidad Pública de Navarra |
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Academica-e. Repositorio Institucional de la Universidad Pública de Navarra |
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