# A Newton method for harmonic mappings in the plane

## TL;DR

This paper introduces a Newton-like iterative method for finding zeros of harmonic mappings in the complex plane, extending classical Newton's method to a broader class of functions with practical implementation and numerical analysis.

## Contribution

It develops a harmonic Newton method for harmonic mappings, including convergence guarantees near poles and critical points, with a Matlab implementation and numerical illustrations.

## Key findings

- Convergence of the method near poles and critical points.
- Effective initial points for guaranteed convergence.
- Numerical experiments demonstrating the method's performance.

## 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 $f = h + \bar{g}$ we construct initial points for which the harmonic Newton iteration is guaranteed to converge. Moreover, we study the number of solutions of $f(z) = \eta$ close to the critical set of $f$ for certain $\eta \in \mathbb{C}$. 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.