An iterative method for conformal mapping

摘要

In this paper we discuss an iterative method for the calculation of the boundary values of the conformal mapping of a simply connected region G onto a region H with smooth boundary. The method is based on a certain Riemann-Hilbert problem. It turns out that this problem is the linearized version of a singular integral equation of the second kind. Hence the method is a Newton method. Whenever the boundaries of G and H are sufficiently smooth, its convergence is locally quadratic.If G is the unit circle, the solution of the linearized problem can be represented explicitly in terms of integral transforms. From this one derives a quadratically-convergent Newton-like numerical method that avoids the numerical solution of systems of linear equations and therefore, in comparison with other methods based on integral equations, is quite economical in terms of computer time and storage requirements.