The computation of zeros of Ahlfors map for multiply connected regions

The relation between the Ahlfors map and Szegö kernel S (z, a) is classical. The Szegö 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 known for a particular family of doubly connected regions and a particular triply connected region. This paper presents a numerical method for computing the zeros of the Ahlfors map of any bounded multiply connected regions with smooth boundaries. The method depends on the values of S (z(t), a), S′(z(t), a) and θ′(t), where θ(t) is the boundary correspondence function of Ahlfors map. A formula is derived for computing S′(z(t), a). An integral equation for θ′(t) is used for finding the zeros of Ahlfors map. The numerical examples presented here demonstrate the method.