The use of singular functions for the approximate conformal mapping of doubly-connected domains

Abstract

Let f be the function which maps conformally a given doubly- connected domain onto a circular annulus. We consider the use of two closely related methods for determining approximations to f of the form fn (z) = z exp, ⎪⎩⎪⎨⎧⎭⎬⎫Σ−(z)uan1jjj where {uj} is a set of basis functions. The two methods are respectively a variational method, based on an extremum property of the function H(z) = f′(z)/f(z) - 1/z, and an orthononnalization method, based on approximating the function H by a finite Fourier series sum. The main purpose of the paper is to consider the use of the two methods for the mapping of domains having sharp corners, where corner singularities occur. We show, by means of numerical examples, that both methods are capable of producing approximations of high accuracy for the mapping of such "difficult" doubly-connected domains. The essential requirement for this is that the basis set {uj} contains singular functions that reflect the asymptotic behaviour of the function H in the neighbourhood of each "singular" corner.