Fast Integration for Cauchy Principal Value Integrals of Oscillatory Kind
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.
cauchy principal value oscillatory convergence hermite interpolation numerical quadrature 65d30 65d32