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.

Airy-type integrals, Numerical quadrature of oscillatory integrals, Numerical integration, Asymptotic approximations, Saddle point analysis, Computing special functions

Gil, A, Segura, J, & Temme, N.M. (2020). Numerical evaluation of Airy-type integrals arising in uniform asymptotic analysis.