Numerical computation of the Schwarz function

TL;DR

This paper presents a method to compute the Schwarz function for analytic arcs using the AAA rational approximation algorithm, enabling solutions to Laplace and PDE problems in the plane.

Contribution

It introduces a novel application of the AAA algorithm to compute the Schwarz function, facilitating analytic continuation and PDE problem solving.

Findings

Successfully computes Schwarz function on analytic arcs

Enables solving Laplace and related PDEs in the plane

Addresses challenges of multi-valuedness away from the arc

Abstract

An analytic function can be continued across an analytic arc $\Gamma$ with the help of the Schwarz function $S(z)$, the analytic function satisfying $S(z) = \bar z$ for $z\in \Gamma$. We show how $S(z)$ can be computed with the AAA algorithm of rational approximation, an operation that is the basis of the AAALS method for solution of Laplace and related PDE problems in the plane. We discuss the challenge of computing $S(z)$ further away from from $\Gamma$, where it becomes multi-valued.