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.
Highlights
A conformal mapping that maps a multiply connected region Ω of connectivity n > 1 onto a unit disk E {w : |w| < 1} is known as the Ahlfors map. It generalizes the Riemann map for a connected region. e Ahlfors map with a base point a ∈ Ω is a n-to-one map
It maps each boundary Γ of Ω corresponding in a one-to-one manner onto the boundary of the unit disk and maps a to the origin [1,2,3]. e Ahlfors map of a multiply connected region has several applications in modern function theory
Unlike [3], the method does not depend on the series representation of the Szegokernel
Summary
A conformal mapping that maps a multiply connected region Ω of connectivity n > 1 onto a unit disk E {w : |w| < 1} is known as the Ahlfors map. E integral equations with the Kerzman–Stein kernel, Neumann kernel, and Szegokernel related to the Ahlfors map for a doubly connected region have been derived in [9]. E second zero of the Ahlfors map for any doubly connected region has been computed numerically in [13] based on the integral equation and Newton iterative method. Is integral equation together with knowledge of intersection points provides another analytical method for finding the second zero of the Ahlfors map for an annulus region.
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