Fast Integration for Cauchy Principal Value Integrals of Oscillatory Kind

Abstract

In this paper we give a simple, but high order and rapid convergence method for computing the Cauchy principal value integrals of the form $\int_{-1}^{1}e^{i\omega x}\frac{f(x)}{x-\tau}dx$ and its error bounds, where f(x) is a given smooth function, ??R + may be large and ?1<?<1. The proposed method is constructed by approximating $(\frac{f(x)-f(\tau)}{x-\tau})^{(s)}$ by using the special Hermite interpolation polynomial, which is a Taylor series. The validity of the method has been demonstrated by the results of several numerical experiments and the comparisons with other methods.