Sinc-Nyström Method for Numerical Solution of One-Dimensional Cauchy Singular Integral Equation Given on a Smooth Arc in the Complex Plane*

We develop a numerical method based on Sine functions to obtain an approximate solution of a one-dimensional Cauchy singular integral equation (CSIE) over an arbitrary, smooth, open arc L of finite length in the complex plane. At the outset, we reduce the CSIE to a Fredholm integral equation of the second kind via a regularization procedure. We then obtain an approximate solution to the Fredholm integral equation by means of Nyström's method based on a Sine quadrature rule. We approximate the matrix and right-hand side of the resulting linear system by an efficient method of computing the Cauchy principal value integrals. The error of an Af-point approximation converges to zero at the rate 0(e~cN ), as N —> oo, provided that the coefficients of the CSIE are analytic in a region D containing the arc L and satisfy a Lipschitz condition in D.