Conformal Mapping via a Density Correspondence for the Double-Layer Potential
Provides a new numerical tool for computing conformal maps, which is important for applications in complex analysis and potential theory, but the improvement over existing methods is not quantified.
The authors derive a representation formula for harmonic and Laurent polynomials using double-layer potential densities, enabling a numerical method for computing conformal maps for both exterior and interior regions. Numerical experiments demonstrate accuracy and broad applicability.
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.