Numerical evaluation of Airy-type integrals arising in uniform asymptotic analysis

2020/01/06 by Amparo Gil, Javier Segura, Gil, Amparo +3
Computer Science · Mathematics · #65D30 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #G.1 #Numerical Analysis (math.NA) #acm:65D30 #cs.NA #math.CA #math.NA #msc:65D30

paper · pdf · doi:10.48550/arxiv.2001.02097

29 pages, 8 figures

arxiv created 2020/01/06 · arxiv updated 2020/01/08

Abstract

We describe a method to evaluate integrals that arise in the asymptotic analysis when two saddle points may be close together. These integrals, which appear in problems from optics, acoustics or quantum mechanics as well as in a wide class of special functions, can be transformed into Airy-type integrals and we use the trapezoidal rule to compute these integrals numerically. The quadrature method, which remains valid when two saddle points coalesce, is illustrated with numerical examples.

Related

• Uniform Asymptotic Methods for Integrals
• Computation of the coefficients appearing in the uniform asymptotic expansions of integrals
• Asymptotic representations for some functions and integrals connected with the Airy function
• Which Saddles Contribute? The South-East Rule for Multidimensional Integrals
• Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals. II. Higher-order transformations