Formation and Analysis of Ahlfors Map Integral Equations for the Regions with Multiple Connectivity

Abstract

This research work elaborates the detailed numerical analysis of the integral equation related to Szegö kernel and Ahlfors map with the assistance of Mathematica 12.0. The Ahlfors map and Szegö kernel are both associated to each other. This is a Fredholm integral equation of second kind via Kerzman-Stein kernel where Szegö kernel is the solution of this integral equation. The solution of integral equation is obtained by transforming the integral equations into matrix form after discretizing. This work presents the computation techniques of Ahlfors map, its zeros, boundary values and the interior points of a region under Ahlfors map. This research construct one new Ahlfors map integral equation with Neumann-type kernel for the regions with multiple connectivity onto a unit disk. The kernel and non-homogeneous term of this equation contain zeros of Ahlfors map. This integral equations is constructed by a non-homogeneous boundary relationship for the regions with multiple connectivity. The new Ahlfors map integral equation is analyzed by parameterizing and discretizing. Discretized integral equations lead to a system of linear equations and give required solution.