Numerical conformal mapping methods based on function conjugation

Abstract

A unifying treatment of methods for computing conformal maps from the unit disk onto a Jordan region is presented. Integral and integro-differential equations (involving the conjugation operator) for the boundary correspondence function are first derived using an arbitrary auxiliary function having certain properties. Then various iteration methods for solving these equations are described in this generality, so that the basic ideas become manifest. Specific methods are then treated as examples of the general theory. Among them are, in particular, the successive conjugation methods of Theodorsen, Melentiev and Kulisch, Timman, and Friberg, the projection method of Bergström, and the Newton methods of Vertgeim, Wegmann, and Hübner (which make use of the easy construction of the solutions of Riemann-Hilbert problems). Many of these methods are treated in greater generality than in the literature. The connections with the methods of Fornberg, Menikoff-Zemach, Chakravarthy-Anderson, and Challis-Burley are also outlined.