Numerical conformal mapping and analytic continuation
Frederic Bisshopp · Quarterly of Applied Mathematics · 1983
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 O ( N log N ) O\left ( {N\log N} \right ) . A similar factoring of the discrete Poisson integral formula allows an explicit inversion of it in O ( N log N ) O\left ( {N\log N} \right ) operations instead of O ( N 3 ) O\left ( {{N^3}} \right ) (N 3 ^{3} ). 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.