Asymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integrals

Ferreira, Chelo (Universidad de Zaragoza) ; López, José L. ; Pérez Sinusía, Ester (Universidad de Zaragoza)

Resumen: We consider the highly oscillatory integral () ∶= ∫ ∞ −∞ (+2+) () for large positive values of , − < ≤ , and positive integers with 1 ≤ ≤ , and () an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when → +∞ for general values of and in terms of elementary functions, and determine the Stokes lines. For ≠ 1, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters and ; the special case =1 requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals Ψ(1, 2,…,) for large values of one of its variables, say , and bounded values of the remaining ones. This family of integrals may be written in the form () for appropriate values of the parameters , and the function (). Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large ||. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al.

Idioma: Inglés

DOI: 10.1111/sapm.12539

Año: 2023

Publicado en: STUDIES IN APPLIED MATHEMATICS 150, 1 (2023), 254-275
ISSN: 0022-2526

Factor impacto JCR: 2.6 (2023)

Categ. JCR: MATHEMATICS, APPLIED rank: 28 / 332 = 0.084 (2023) - Q1 - T1

Factor impacto CITESCORE: 4.3 - Applied Mathematics (Q1)

Factor impacto SCIMAGO: 1.009 - Applied Mathematics (Q1)

Tipo y forma: Artículo (Versión definitiva)

Área (Departamento): Área Matemática Aplicada (Dpto. Matemática Aplicada)