The computation of zeros of Ahlfors map for doubly connected regions

Abstract

The Ahlfors map and Szegö kernel are both classically related to each other. Ahlfors map can be computed using Szege kernel without relying on the zeros of Ahlfors map. The Szegö kernel is a solution of a Fredholm integral equation of the second kind with the Kerzman­Stein 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 Szegö kernel, its derivative and the derivative of boundary correspondence function of Ahlfors map. Using combination of Nyström method, GMRES method, fast multiple method and Newton's method,the numerical examples presented here prove the effectiveness of the proposed method.