Application of the Cauchy integral approach to singular and highly oscillatory integrals

Abstract

This paper presents a method that is based on the sum of line integrals for fast computation of singular and highly oscillatory integrals ${\int }_c^{d}G(x){\mathrm{e}}^{\mathrm{i}\mu (x-c{)}^k}\mathrm{d}x,-\mathrm{\infty }>c>d>\mathrm{\infty }$, and ${\int }_{-1}^{1}f(x)H_l(x){\mathrm{e}}^{\mathrm{i}\mu x}\mathrm{d}x,l=1,2,3$. Where G and f are non-oscillatory sufficiently smooth functions on the interval of integration. $H_l$ is a product of singular factors and $\mu \gg 1$ is an oscillatory parameter. The computation of these integrals requires f and G to be analytic in a large complex region C accommodating the interval of integration. The integrals are changed into a problem of integrals on $[0,\mathrm{\infty })$; which are later computed using the generalized Gauss-Laguerre rule or by the construction of Gauss rules relative to a Freud weights function $e^{-x^k}$ with k positive. MATHEMATICA programming code, algorithms and illustrative numerical examples are provided to test the efficiency of the presented experiments.

Keywords:

• Highly oscillatory integrals
• Gauss quadrature rules
• Numerical steepest descent method
• Gauss–Laguerre quadrature rule
• Chebyshev algorithm

2010 Mathematics Subject Classification:

• 65D30

Disclosure statement

No potential conflict of interest was reported by the author(s).