Integral equation for the Ahlfors map on multiply connected regions

Abstract

This paper presents a new boundary integral equation with ɵ’ the adjoint Neumann kernel associated with where ɵ’ is the boundary correspondence function of Ahlfors map of a bounded multiply connected region onto a unit disk. The proposed boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a multiply connected region. The integral equation is solved numerically for ɵ’ using combination of Nystrom method, GMRES method, and fast multiple method. From the computed values of ɵ’ we solve for the boundary correspondence function ɵ’ which then gives the Ahlfors map. The numerical examples presented here prove the effectiveness of the proposed method