Analytical solution for finding the second zero of the Ahlfors map for an annulus region

Abstract

The Ahlfors map is a conformal mapping function that maps a multiply connected region onto a unit disk. It can be written in terms of the Szegö kernel and the Garabedian kernel. In general, a zero of the Ahlfors map can be freely prescribed in a multiply connected region. The remaining zeros are the zeros of the Szegö kernel. For an annulus region, it is known that the second zero of the Ahlfors map can be computed analytically based on the series representation of the Szegö kernel. This paper presents another analytical method for finding the second zero of the Ahlfors map for an annulus region without using the series approach but using a boundary integral equation and knowledge of intersection points.