Efficient numerical methods for Cauchy principal value integrals with highly oscillatory integrands

This paper focus on the numerical evaluation of the Cauchy principal value integrals with oscillatory integrands [inline-graphic not available: see fulltext] where α, β > − 1,− 1 < τ < 1. For the case f is analytic in a sufficiently large region containing [− 1,1], the integrals can be transformed into the problems of integrating two line integrals, the integrands of which do not oscillate and decay exponentially fast, and thus can be computed by using Gaussian quadrature rules. For the smooth function f, a method is constructed by interpolating f at Clenshaw–Curtis points and the singular point τ, based on the fast computation of modified moments. Error bounds of two proposed methods are both presented. In addition, several numerical examples are given to illustrate the efficiency and accuracy of proposed methods.