An integral equation method for conformal mapping of doubly and multiply connected regions via the kerzaman-stein and neumann kernels

Abstract

This research develops some 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. The integral equations are constructed from a boundary relationship satis¯ed 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 non-linear 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, we can use the Cauchy's integral formula 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 su±ciently smooth.