Computable Caratheodory Theory

TL;DR

This paper develops a constructive version of Carathéodory Theory for the boundary extension of the Riemann map, providing explicit complexity bounds for computational aspects in classical complex analysis.

Contribution

It introduces a constructive approach to Carathéodory Theory with explicit complexity bounds, advancing computational methods in conformal mapping.

Findings

Provides explicit complexity bounds for boundary extension computations

Develops a constructive version of Carathéodory Theory

Enhances computational tools for conformal mappings

Abstract

Conformal Riemann mapping of the unit disk onto a simply-connected domain $W$ is a central object of study in classical Complex Analysis. The first complete proof of the Riemann Mapping Theorem given by P. Koebe in 1912 is constructive, and theoretical aspects of computing the Riemann map have been extensively studied since. Carath{\'e}odory Theory describes the boundary extension of the Riemann map. In this paper we develop its constructive version with explicit complexity bounds.