The Computation of Zeros of Ahlfors Map for 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.