An improved algorithm for the evaluation of Cauchy principal value integrals of oscillatory functions and its application

A new interpolatory-type quadrature rule is proposed for the numerical evaluation of Cauchy principal value integrals of oscillatory kind ⨍−11f(x)x−τeiωxdx, where τ∈(−1,1)τ∈(−1,1). The method is based on an interpolatory procedure at Clenshaw–Curtis points and the singular point, and the fast computation of the modified moments with Cauchy type singularity. Based on this result, a new method is presented for the computation of the oscillatory integrals with logarithmic singularities too. These methods enjoy fast implementation and high accuracy. Convergence rates on ωω are also provided. Numerical examples support the theoretical analyses.