A Newton method for harmonic mappings in the plane

Abstract

We present an iterative root finding method for harmonic mappings in the complex plane, which is a generalization of Newton's method for analytic functions. The complex formulation of the method allows an analysis in a complex variables spirit. For zeros close to poles of f = h + g we construct initial points for which the harmonic Newton iteration is guaranteed to converge. Moreover, we study the number of solutions of f (z) = η close to the critical set of f for certain η ∈ C. We provide a Matlab implementation of the method, and illustrate our results with several examples and numerical experiments, including phase plots and plots of the basins of attraction.