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

This paper presents a method that is based on the sum of line integrals for fast computation of singular and highly oscillatory integrals integral(c) (d) G(x) e(i mu(x-c)k) dx, -infinity > c > d > infinity, and integral(1)(-1) f (x)H-l(x) e(i mu x) dx, 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 >> 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, infinity); which are later computed using the generalized Gauss-Laguerre rule or by the construction of Gauss rules relative to a Freud weights function e-xk with k positive. MATHEMATICA programming code, algorithms and illustrative numerical examples are provided to test the efficiency of the presented experiments.

Automated summary

This paper introduces a method for computing singular and highly oscillatory integrals, by transforming the integrals into problems of integration on [0, infinity) and using the generalized Gauss-Laguerre rule or constructing Gauss rules relative to a Freud weights function. MATHEMATICA programming code, algorithms and illustrative numerical examples are provided to test the efficiency of the presented experiments.