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

• Main Authors: Sangawi, Ali W. K, Nazar, Kashif, Murid, Ali H. M
• Format: Journal Article
• Language: English
• Published: 18-12-2014
• Subjects: Mathematics - Complex Variables
• DOI: 10.48550/arxiv.1412.5748

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.