Comparisons of the Computation of Zeros of Ahlfors Map in Bounded Doubly Connected Regions

Abstract

This thesis compares and analyzes various methods for determining the zeros of the Ahlfors map in bounded doubly connected regions. The Riemann mapping theorem’s extension to multiply connected domains, known as the Ahlfors map, is crucial for complex analysis and has important uses in engineering and science fields like fluid dynamics. The two special points, or zeros, of the Ahlfors map in doubly connected regions can be found using a variety of methods described in the literature. The Ahlfors map is a fundamental mathematical tool that enables us to project complex structures onto a unit disk while preserving the same angles. The second zero of the Ahlfors map is more difficult to locate than the first. Several mathematical and numerical techniques, including boundary integral equations, and specific kernels like the Szego˝ and Neumann kernels, are used and compared in the thesis. The research compares these methods to see which ones are more accurate, faster, and more stable when applied to different types of doubly connected regions. The results help identifies the best approaches for calculating the Ahlfors map’s zeros and can be useful in practical fields like fluid dynamics, electricity, and image processing. This work also helps future studies on more complex shapes and higher connectivity regions.