Numerical computation of the Schwarz function (2501.00898v1)

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.

• The paper introduces a novel method for the numerical computation of the Schwarz function using the AAA algorithm, providing a reliable approach where previous methods were lacking.
• Applying the AAA algorithm effectively approximates the multi-valued Schwarz function, with poles revealing structures near branch cuts essential for analytic continuation across complex arcs.
• This numerical method has significant implications for solving partial differential equations like Laplace and Helmholtz equations in complex domains and advancing boundary-value problem solutions.

Numerical Computation of the Schwarz Function

The paper discusses a novel approach for the numerical computation of the Schwarz function, an important complex analytic function that extends other analytic functions across an analytic arc. Although the concept of the Schwarz function is well-documented in mathematical literature, its practical use has been limited by the absence of reliable numerical methods for its computation. This work utilizes the AAA algorithm for rational approximation to provide a computational method that addresses this gap.

Key Contributions

1. Approach Using AAA Algorithm: The algorithm, described as a "step change" from previous methods, offers near-optimal rational approximations by employing a barycentric representation and a greedy process that iteratively selects coefficients through linear least-squares problems. The AAA algorithm enables the approximation of complex functions to high accuracy and demonstrates this capability in the computation of the Schwarz function.
2. Effective Approximation: Applying the AAA algorithm to the computation of the Schwarz function, poles manifest in regions approximating branch cuts, highlighting the multi-valued nature of the Schwarz function when considered away from $\Gamma$. This property is crucial for its ability to facilitate analytic continuation in two-dimensional domains.
3. Generalization of the Schwarz Reflection Principle: The paper extends the Schwarz reflection principle for real lines to more general analytic curves. This generalization leverages the properties of the Schwarz function to achieve reflection across complex analytic arcs, enhancing its applicability in various problem settings.
4. Applications to PDEs: The methodology promises significant implications for solving partial differential equations (PDEs), specifically the Laplace and Helmholtz equations, in complex domains. The ability to numerically approximate the Schwarz function can advance boundary-value problem solutions through methods like the AAALS (AAA-least squares) method.

Numerical Results and Implications

The paper provides several numerical examples that illustrate the effectiveness of the AAA algorithm in approximating Schwarz functions for different types of curves, including ellipses and more complex geometries with singularities. The accuracy of these approximations is substantiated by comparing the reflections derived from the approximated Schwarz functions with their analytically known counterparts or by assessing the preservation of analytic properties.

These results imply that the computational method has robust potential for applications that require the analytic continuation of functions, which is critical in fluid dynamics problems, electromagnetic theory, and other areas involving PDEs. The research points to the promise of rational approximation techniques in scenarios where traditional polynomial approximations may fail due to domain characteristics or the presence of singularities.

Future Directions

The paper leaves room for further inquiry into the utilization of computed Schwarz functions in more complex PDE scenarios and other analytic continuation challenges. The interaction between high-accuracy numerical methods like the AAA algorithm and intricate domain geometries necessitates further exploration, particularly concerning the optimization of the choice between branches in multivalued function scenarios.

Moreover, the integration of this approach with existing numerical PDE solvers and frameworks could lead to enhanced computational efficiency and accuracy in applied mathematics, physics, and engineering fields, aiding the eventual transition from theoretical formulations to practical computations in intricate geometries. The work underscores the need for advanced software implementations and optimization techniques to harness the full potential of the proposed method.

In summary, this paper presents a clear, technically rigorous advancement in the numerical computation of the Schwarz function, supported by the innovative application of the AAA algorithm, which may profoundly impact various fields requiring complex analytic continuation.