Conformal Mapping via a Density Correspondence for the Double-Layer Potential

TL;DR

This paper introduces a novel numerical method for computing conformal maps using a density correspondence for the double-layer potential, applicable to both interior and exterior regions of bounded domains, supported by analysis and experiments.

Contribution

It provides a new representation formula for harmonic polynomials via densities of the double-layer potential, enabling efficient conformal mapping computations.

Findings

Method achieves high accuracy in numerical experiments

Applicable to both interior and exterior conformal mapping

Supported by theoretical analysis and numerical validation

Abstract

We derive a representation formula for harmonic polynomials and Laurent polynomials in terms of densities of the double-layer potential on bounded piecewise smooth and simply connected domains. From this result, we obtain a method for the numerical computation of conformal maps that applies to both exterior and interior regions. We present analysis and numerical experiments supporting the accuracy and broad applicability of the method.