A simplification of the stationary phase method: application to the Anger and Weber functions
Ver/
Fecha
2017Autor
Versión
Acceso abierto / Sarbide irekia
Tipo
Artículo / Artikulua
Versión
Versión aceptada / Onetsi den bertsioa
Impacto
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nodoi-noplumx
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Resumen
The main difficulty in the practical use of the stationary phase method in asymptotic expansions of
integrals is originated by a change of variables. The coefficients of the asymptotic expansion are the coefficients of
the Taylor expansion of a certain function implicitly defined by that change of variables. In general, this function is
not explicitly known, and then the computation of those c ...
[++]
The main difficulty in the practical use of the stationary phase method in asymptotic expansions of
integrals is originated by a change of variables. The coefficients of the asymptotic expansion are the coefficients of
the Taylor expansion of a certain function implicitly defined by that change of variables. In general, this function is
not explicitly known, and then the computation of those coefficients is cumbersome. Using the factorization of the
exponential factor used in previous works of [Tricomi, 1950], [Erdélyi and Wyman, 1963], and [Dingle, 1973], we
obtain a variant of the method that avoids that change of variables and simplifies the computations. On the one hand,
the calculation of the coefficients of the asymptotic expansion is remarkably simpler and explicit. On the other hand,
the asymptotic sequence is as simple as in the standard stationary phase method: inverse powers of the asymptotic
variable. New asymptotic expansions of the Anger and Weber functions Jλx(x) and Eλx(x) for large positive x and
real parameter λ 6= 0 are given as an illustration. [--]
Materias
Asymptotic expansions,
Oscillatory integrals,
Method of the stationary phase,
Anger and Weber functions
Editor
Kent State University Johann Radon Institute (RICAM)
Publicado en
Electronic Transactions on Numerical Analysis, Volume 46, pp. 148–161, 2017
Departamento
Universidad Pública de Navarra. Departamento de Ingeniería Matemática e Informática /
Nafarroako Unibertsitate Publikoa. Matematika eta Informatika Ingeniaritza Saila /
Universidad Pública de Navarra/Nafarroako Unibertsitate Publikoa. Institute for Advanced Materials and Mathematics - INAMAT2
Entidades Financiadoras
This research was supported by the Spanish Ministry of Economía y Competitividad, project MTM2014-52859-P. The Universidad Pública de Navarra is acknowledged by its financial support.