Boundary integral equations approach for numerical conformal mapping of multiply connected regions

Abstract

Several integral equations involving the Kerzman-Stein and the Neumann kernels for conformal mapping of multiply connected regions onto an annulus with circular slits and onto a disk with circular slits are presented. The theoretical development is based on the boundary integral equation for conformal mapping of doubly connected region. The integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. The boundary integral equations involve the unknown parameter radii. For numerical experiments, discretizing each of the integral equations leads to a system of nonlinear equations. Together with some normalizing conditions, a unique solution to the system is then computed by means of an optimization method. Once the boundary values of the mapping function are calculated, the Cauchy’s integral formula has been used to determine the mapping function in the interior of the region. Typical examples for some test regions show that numerical results of high accuracy can be obtained for the conformal mapping problem when the boundaries are sufficiently smooth.