The asymptotic expansion of the swallowtail integral in the highly oscillatory region
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Fecha
2018Versión
Acceso abierto / Sarbide irekia
Tipo
Artículo / Artikulua
Versión
Versión aceptada / Onetsi den bertsioa
Impacto
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10.1016/j.amc.2018.07.008
Resumen
The mathematical models of many short wavelength phenomena, specially wave propagation and optical diffraction,
contain, as a basic ingredient, oscillatory integrals with several nearly coincident stationary phase or saddle points. The
uniform approximation of those integrals can be expressed in terms of certain canonical integrals and their derivatives
[2,16]. The importance of these canonica ...
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The mathematical models of many short wavelength phenomena, specially wave propagation and optical diffraction,
contain, as a basic ingredient, oscillatory integrals with several nearly coincident stationary phase or saddle points. The
uniform approximation of those integrals can be expressed in terms of certain canonical integrals and their derivatives
[2,16]. The importance of these canonical diffraction integrals is stressed in [14] by means of the following sentence: The
role played by these canonical diffraction integrals in the analysis of caustic wave fields is analogous to that played by complex
exponentials in plane wave theory. Apart from their mathematical importance in the uniform asymptotic approximation of oscillatory integrals [12], the
canonical diffraction integrals have physical applications in the description of surface gravity waves [11], [17], bifurcation sets, optics, quantum mechanics, chemical physics [4] and acoustics (see [1], Section 36.14 and references there in). To our
knowledge, the first application of this family of integrals traces back to the description of the disturbances on a water surface produced, for example, by a traveling ship. These disturbances form a familiar pattern of bow and stern waves
which was first explained mathematically by Lord Kelvin [10] using these integrals. [--]
Materias
Swallowtail integral,
Asymptotic expansions,
Modified saddle point method
Editor
Elsevier
Publicado en
Applied Mathematics and Computation 339 (2018) 837–845
Departamento
Universidad Pública de Navarra. Departamento de Ingeniería Matemática e Informática /
Nafarroako Unibertsitate Publikoa. Matematika eta Informatika Ingeniaritza Saila
Versión del editor
Entidades Financiadoras
This research was supported by the Ministerio de Economía y Competitividad (MTM2014-52859) and the Universidad Pública de Navarra.