Numerical computation of the Schwarz function

TL;DR

The paper presents a numerical framework for computing the Schwarz function $S(z)$ of an analytic arc $\Gamma$ using the AAA rational approximation method, enabling analytic continuation and related PDE techniques in the plane. By sampling $\Gamma$ and applying AAA to the data $F=\overline{Z}$, it yields a rational approximant $r$ that matches $S$ on $\Gamma$ with high accuracy, while the poles of $r$ reveal branch-cut structures and multivaluedness away from $\Gamma$. The work demonstrates rapid, accurate computation on various curves, including singular and nonconvex geometries, and analyzes how $r$ captures multiple Schwarz branches and how far the approximation remains reliable. The discussion highlights the AAALS framework for solving 2D Dirichlet problems and indicates potential extensions to Helmholtz and Stokes-type problems, with the Schwarz function playing a central role in understanding convergence behavior near concave boundaries.

Abstract

An analytic function can be continued across an analytic arc $Γ$ with the help of the Schwarz function $S(z)$, the analytic function satisfying $S(z) = \bar z$ for $z\in Γ$. 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 $Γ$, where it becomes multi-valued.

Figures (5)

• Figure 1: On the left, the Schwarz reflection principle (Theorem 1) extends an analytic function that takes real values on a real interval across the interval from one half-plane to the other. On the right, the generalization by the Schwarz function $S(z)$ for reflection of an analytic function across an analytic arc $\Gamma$, again assuming it takes real values on $\Gamma$.
• Figure 2: In the first row, poles (blue) of AAA approximations to $S(z)$ for an ellipse and a half-ellipse. These computations took less than $0.01$ s on a laptop. Both cases show a string of poles approximating a branch cut extending from $-1$ to $1$. In each image three orange points $z$ have been selected and their reflections $\overline{r(z)}\approx\overline{S(z)}$ shown as green circles. In the second row, error indicators $(\ref{['measure']})$ of the approximations $r(z)\approx S(z)$. In the dark green region the accuracy is better than $10^{-8}$, and in the light green region it is better than $10^{-1}$.
• Figure 3: Repetition of Figure \ref{['figellipse']} for a different pair of curves $\Gamma$.
• Figure 4: Four more examples of numerically computed Schwarz functions. In the third image $\Gamma$ has a singularity at $z=0$, and in the fourth one, there are six singularities at the six corners. In both of these cases there are several mathematically independent branches of $S(z)$, which the rational approximations separate by strings of poles approximating branch cuts. For the final image the tolerance has been loosened to $10^{-8}$ and the dark green region shows accuracy $10^{-5}$ rather than the usual $10^{-8}$.
• Figure 5: Repetition of the bottom row of Figure \ref{['figellipse']} with (\ref{['measure']}) replaced by (\ref{['measure2']}), so that the plot indicates deviation at each point from the closer of the two branches of the exact Schwarz function $S(z)$ of $(\ref{['SE']})$, and with colors changed from green to blue. Except along the approximate branch cut, $r$ matches one or the other branch of $S(z)$ with good accuracy everywhere. The dark and light regions extend distances from the origin about $4000$ and $1.5\times 10^7$ on the left, $6$ and $5000$ on the right.

Theorems & Definitions (3)

• Theorem 1
• Definition 1
• Theorem 2