The Computation of Zeros of Ahlfors Map for Doubly Connected~Regions

TL;DR

This paper introduces a numerical method to compute the zeros of the Ahlfors map for any bounded doubly connected region, leveraging the Szeg"o kernel and boundary correspondence functions.

Contribution

It develops a novel numerical approach for finding Ahlfors map zeros in doubly connected regions, extending beyond the annulus case.

Findings

Method effectively computes zeros in various regions.

Numerical examples demonstrate high accuracy.

The approach is applicable to general doubly connected regions.

Abstract

The relation between the Ahlfors map and Szeg\"o kernel S(z, a) is classical. The Szeg\"o kernel is a solution of a Fredholm integral equation of the second kind with the Kerzman-Stein kernel. The exact zeros of the Ahlfors map are unknown except for the annulus region. This paper presents a numerical method for computing the zeros of the Ahlfors map of any bounded doubly connected region. The method depends on the values of S(z(t),a), S'(z(t),a) and \theta'(t) where \theta(t) is the boundary correspondence function of Ahlfors map. A formula is derived for computing S'(z(t),a). An integral equation is constructed for solving \theta'(t). The numerical examples presented here prove the effectiveness of the proposed method.