A Constructive Method for Numerically Computing Conformal Mappings for Gearlike Domains

The Riemann mapping theorem asserts that the open unit disk D = {z| |z| < 1} is conformally equivalent to each simply connected domain G in the complex plane, whose boundary consists of at least two points, i.e., there exists a function f , analytic and univalent function on D , such that f maps D onto G . More precisely, if do is an arbitrary point in D and go is an arbitrary point in G, then the Riemann mapping theorem asserts that there exists a unique conformal mapping f of D onto G such that f(do) = go and f ′(do) > 0. If the boundary of G is piece-wise analytic and g1 is a point on the boundary of G, then the uniqueness assertion of the Riemann mapping theorem can be reformulated alternately as the statement that there exists a unique conformal mapping f of D onto G such that f(do) = go and f(1) = g1. The problem of constructing the explicit conformal mapping guaranteed by the Riemann mapping theorem is usually difficult, even numerically. There are constructive proofs [He] of the Riemann mapping theorem, but, because of their general nature, they often converge only slowly to the desired solution. For polygonal domains, i.e., simply connected domains bounded by segments of straight lines, there is a well-known representation formula, called the Schwarz-Christoffel formula (see [Ne], p. 189), which the conformal mapping functions must satisfy. The difficulty with using the Schwarz-Christoffel formula to construct the conformal mapping for an explicit polygonal domain has always been that the formula contains unknown parameters, called the accessory parameters, which have to be determined before the mapping func-