The swallowtail integral in the highly oscillatory region III
Fecha
2021Versión
Acceso abierto / Sarbide irekia
Tipo
Artículo / Artikulua
Versión
Versión aceptada / Onetsi den bertsioa
Identificador del proyecto
Impacto
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10.1080/17476933.2020.1868447
Resumen
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 saddl ...
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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. [--]
Materias
Swallowtail integral,
Asymptotic expansions,
Modified saddle point method
Editor
Taylor & Francis
Publicado en
Complex Variables and Elliptic Equations
Departamento
Universidad Pública de Navarra. Departamento de Estadística, Informática y Matemáticas /
Nafarroako Unibertsitate Publikoa. Estatistika, Informatika eta Matematika Saila /
Universidad Pública de Navarra/Nafarroako Unibertsitate Publikoa. Institute for Advanced Materials and Mathematics - INAMAT2
Versión del editor
Entidades Financiadoras
This research was supported by the Ministerio de Economía y Competitividad (MTM2017-83490-P) and the Universidad Pública de Navarra.