Finding the zeros of the ahlfors map for smooth multiply connected regions
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Date
2020
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Publisher
Universiti Teknologi Malaysia
Abstract
The Ahlfors map is a conformal mapping function that maps a multiply connected region onto a unit disk and can be written in terms of the Szego kernel. The Szego kernel is known to satisfy the Kerzman-Stein integral equation. Specifically for an annulus region, it is known that the second zero of the Ahlfors map can be obtained analytically based on the series representation of the Szego kernel obtained from orthonormal basis. This research provides a new series form for the Szego kernel for the annulus region and is shown to be equivalent to the existing series. The series is obtained by solving the Kerzman-Stein integral equation using the Adomian decomposition method. This new series gives faster convergence than the existing series. In general, a zero of the Ahlfors map can be freely prescribed in a multiply connected region and the remaining zeros are unknown except for an annulus region and for a special triply connected region. It is known that the zeros of the Ahlfors map for a multiply connected region satisfy a system of nonlinear equations involving an integral containing the Szego kernel and the derivative of the Szego kernel. However, there is no numerical computation that has been done previously to solve the system of nonlinear equations for computing the zeros of the Ahlfors map. A boundary integral equation with Neumann kernel related to the Ahlfors map has been derived previously by using a boundary relationship satisfied by the Ahlfors map. This integral equation however consists of an imaginary part containing the zeros of the Ahlfors map thus making it difficult for computing the zeros of the Ahlfors map. This research presents a new integral equation related to the Ahlforsmap that is similar to the existing integral equation but without taking the imaginary part. The new integral equation is then used to compute the zeros of the Ahlforsmap for smooth multiply connected regions. Some numerical implementations for computing the zeros of the Ahlfors map using the new integral equation and existing formulas based on a system of nonlinear equations and boundary integral equation with Neumann kernel are also presented and compared. This research has shown that the accuracies of the obtained zeros of the Ahlfors map using the new integral equation and existing formulas are comparable. However, the numerical implementations for computing the zeros using the new integral equation is easier than the boundary integral equation with Neumann kernel. It is also shown that the computation times (in seconds) for computing the zeros of the Ahlfors map using the new integral equation and existing formula based on a system of nonlinear equations is faster than the boundary integral equation with Neumann kernel. Another analytical method based on the new integral equation and knowledge of geometrical properties is also derived to find the second zero of the Ahlfors map for an annulus region. This method does not depend on the series representation of the Szego kernel for an annulus region.
Description
Thesis (PhD. (Chemistry))
Keywords
Mathematical geography, Map projection