A Boundary Integral Equation Related to the Ahlfors Map

The Ahlfors map is a mapping function that maps a multiply connected region onto a unit disk. This paper presents a new boundary integral equation related to the Ahlfors map of a bounded multiply connected region. The boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a multiply connected region.

Analytical Solution for Finding the Second Zero of the Ahlfors Map for an Annulus Region

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.

Integral Equation for the Ahlfors Map on Multiply Connected Regions

This paper presents a new boundary integral equation with the adjoint Neumann kernel associated with where 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 using combination of Nystrom method, GMRES method, and fast multiple method. From the computed values of we solve for the boundary correspondence function which then gives the Ahlfors map. The numerical examples presented here prove the effectiveness of the proposed method.