Fast conformal mapping of an ellipse to a simply connected region

The iterative method of Wegmann (1978) for conformal mapping of a region E to a region G is reformulated in terms of a conjugation operator. If the boundary of G is sufficiently smooth, the method converges quadratically in a Sobolev space W. To overcome the crowding an elongated canonical region E should be chosen if the image region G is elongated. For ellipses E the operator of conjugation can be expressed in a very simple way in terms of Fourier series. Numerically, this can be performed very efficiently by fast Fourier transform (FFT). calculation is done on a grid with N points and if only the first m Fourier terms are retained, then for fixed m this smoothed numerical method converges if the initial approximation is sufficiently close to the solution and N is large enough. The limit function approximates the parameter mapping function. An error estimate is given. For regions with analytic boundaries the error is bounded by Cpm for all m with some constant p < 1. These results increase the flexibility and applicability of the method appreciably. Several examples are discussed in detail and some numerical experiments are reported.