A Numerical Method for Conformal Mappings

A numerical technique is presented for calculating the Taylor coefficients of the analytic function which maps the unit circle onto a region bounded by any smooth simply connected curve. The method involves a quadratically convergent outer iteration and a super-linearly convergent inner iteration. If N complex points are distributed equidistantly around the periphery of the unit circle, their images on the edge of the mapped region, together with approximations for the N / 2 first Taylor coefficients, are obtained in O(Nlog N) operations. A calculation of time-dependent waves on deep water is discussed as an example of the potential applications of the method.