A Numerical Method for Locating the Zeros of Ahlfors Map for Doubly Connected Regions

Abstract

The Ahlfors map of an -connected region is a -to-one map from the region onto the unit disk. The Ahlfors map being -to-one map has zeros. Previously, the exact zeros of the Ahlfors map are known only for the annulus region. The zeros of the Ahlfors map for general bounded doubly connected regions has been unknown for many years. 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 Szeg kernel, its derivative and the derivative of boundary correspondence function of the Ahlfors map. The Ahlfors map and Szeg kernel are both classically related to each other. Ahlfors map can be computed using Szeg kernel without relying on the zeros of Ahlfors map. The Szeg kernel is a solution of a Fredholm integral equation of the second kind with the Kerzman-Stein kernel. The numerical examples presented here prove the effectiveness of the proposed method.