Methods and comparisons for computing the zeros of the Ahlfors map for doubly connected regions

Abstract

A conformal mapping function that maps a multiply connected region onto a unit disk is known as the Ahlfors map. It reduces to the Riemann map for a simply connected region. The Ahlfors map can be expressed in terms of the Szego kernel. The Szego kernel is a unique solution to a second-kind Fredholm integral equation. The Ahlfors map and the Szego kernel have the same zeros. There is no known explicit formula for the zeros of the Ahlfors map for doubly connected regions except for the annulus region. For general doubly connected regions, the zeros of the Ahlfors map must be computed numerically. This paper shows how to determine the zeros of the Ahlfors map numerically for any doubly connected regions with smooth boundaries. The numerical examples and comparisons are also presented.