Analysis of the Formulations and Continuities of Kernels of Integral Equations for Ahlfors Map

Abstract

This study aims to examine and analyze the boundary integral equations of the Ahlfors map specifically for multiply connected regions. The Ahlfors map, known for mapping the boundary of a domain to the unit circle, helps in understanding the domain’s structure. Our goal is to rederive boundary integral equations that are more effective in determining the Ahlfors map and its unknown zeros. Identifying these zeros is crucial as they provide important information about the map’s behavior and the domain’s geometry. By develop- ing these new equations, we hope to address existing challenges and solve long-standing problems in the field, ultimately advancing mathematical understanding and applications of the Ahlfors map in complex analysis and related disciplines.