The Green's function and the Ahlfors map

TL;DR

This paper extends the classical Green's function representation from simply connected to finitely connected planar domains using Ahlfors mappings, providing a new analytical tool for complex analysis.

Contribution

It introduces a method to express Green's functions of multiply connected domains via Ahlfors mappings, generalizing the Riemann mapping approach.

Findings

Green's function expressed in terms of Ahlfors mapping

Provides a new analytical framework for multiply connected domains

Bridges classical and modern complex analysis techniques

Abstract

The classical Green's function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green's function of a finitely connected domain in the plane in terms of a single Ahlfors mapping of the domain, which is a proper holomorphic mapping of the domain onto the unit disc that is the analogue of the Riemann map in the multiply connected setting.