Numerical conformal mapping via Chebyshev weighted solutions of Symm's integral equation

Abstract A numerical method is described for the conformal mapping of simply connected domains with piecewise analytic boundary. The method is based on the first-kind integral equation formulation of Symm (1966). On each component analytic arc of the boundary, the end point singularities of the unknown source density σ are annihilated by introducing the classical Chebyshev weight w, so that σ/w may be approximated by a finite Chebyshev polynomial series. The coefficients in these series are determined by collocation. The method, which provides a problem-independent treatment of end point singularities, has the advantages that all nonsingular integrals may be efficiently computed via the FFT and singular integrals are known in simple and exact form. Numerical examples illustrate the effectiveness of the method and also provide experimental confirmation of the partial error analysis of the authors (forthcoming paper).