Numerical conformal mapping and analytic continuation
A numerical method for determination of least-square approximations of an arbitrary complex mapping function is derived here and implemented with fast Fourier transforms (FFTs). An essential feature of the method is the factoring of a discrete Hilbert transform in a pair of Fourier transforms in order to reduce the operation count of the longest computation to 0(N log N). A similar factoring of the discrete Poisson integral formula allows an explicit inversion of it in 0(N log N) operations instead of 0(N3). The resulting scheme for analytic continuation appears to be considerably more reliable than the evaluation of polynomials. Examples are treated, and APL implementations of algorithms are provided.