Numerical techniques for conformal mapping onto a rectangle

This paper is concerned with the problem of determining approximations to the function F which maps conformally a simply-connected domain Ω onto a rectangle R, so that four specified points on ∂Ω are mapped respectively onto the four vertices of R. In particular, we study the following two classes of methods for the mapping of domains of the form Ω≔ {z = x + iy:00 < x < 1, τ 1(x) < y < τ 2(x)} . (i) Methods which approximate F: Ω → R by F ̃ = S ∘ F ̃ , where F̃ is an approximation to the conformal map of Ω onto the unit disc, and S is a simple Schwarz-Christoffel transformation. (ii) Methods based on approximating the conformal map of a certain symmetric doubly-connected domain onto a circular annulus.