A new interpolatory type quadrature rule for weighted Cauchy principal value integrals

Abstract

This paper presents a new interpolatory type quadrature rule for approximating the weighted Cauchy principal value integrals integral(-1)(1)(1 - t(2))(lambda-1/2)f(t)/(t - c)dt where - 1/2 < lambda < 1. We prove that the rule has almost optimal stability property behaving in the form O(K log n + L), where K and L are constants depending only on c. Also, when f(t) possesses continuous derivatives up to order p >= 0 and the derivative f(P)(t) satisfies Holder continuity of order p, we obtain that the rule has the convergence rate of O((A + B log n + n(2v))n(-p-rho)), where v is as small as we like and A and B are constants depending on c.