On the numerical evaluation of Cauchy transforms (Q1315224)

The problem of reconstructing the function defined by the Cauchy principal value of another function arises in many areas of applied mathematics. In this paper the authors consider simple methods for the reconstruction of the Cauchy transform over a curve when an explicit parametrization of the latter is not provided. The methods consist of replacing the parametrization of the curve by piecewise polynomial interpolation followed by the use of Newton-Cotes type formulae for the integration. The order of convergence of the resulting quadrature is higher than would be expected on the basis of considerations involving just interpolating theory, provided that the Cauchy transform is evaluated at known nodes on the curve. These results also allow the calculation of the Cauchy transform at other points with the same accuracy if the scheme is followed by an interpolatory formula of sufficiently high accuracy. The investigation is a preliminary study to construct a reliable method for the calculation of the transformed function.