Numerical conformal mapping via the Bergman kernel using Fourier method

Abstract

The Szego kernel and the Bergman kernel of a simply connected region in the complex plane are kernel functions which are related to the Riemann mapping function. An effcient method based on the Kerzman-Stein-Trummer integral equation for computing the Szego kernel has been known since 1986. In 1997, integral equation for the Bergman kernel which can be used effectively for numerical conformal mapping has also been established. Both of these integral equations have been solved by means of Nystrom’s method. Our subject of study is based on integral equation for Bergman kernel, where we had solved this integral equation by means of Fourier method. Since integral equation for Bergman kernel has not yet been solved using Fourier method, the numerical results can also be used to compare with those obtained from Nystrom’s method. As a result, Fourier method is capable to produce approximations of comparable accuracy to the Nystrom’s method; where these approximations are also suitable for numerical conformal mapping.