On some relations among numerical conformal mapping methods

摘要

Gaier and Gutknecht have shown that many numerical methods for producing the conformal map from the unit disk to a simply connected region share a common theoretical basis as solutions of nonlinear integral equations arising from the Hilbert transform of a function of the boundary correspondence. We give a brief presentation of this classification and extend it somewhat to include some equations for the inverse correspondence, such as those of Menikoff and Zemach, Noble, and Schwarz-Christoffel. The use of explicit maps and the method of Bisshopp are also brought into this framework. An example illustrating the use of explicit maps is given.