Conformal equivalence of analytic functions on compact sets

TL;DR

This paper provides a geometric proof demonstrating that any analytic function on a Jordan domain can be represented as a polynomial composed with an injective analytic map, establishing a conformal equivalence.

Contribution

It introduces a geometric proof of the conformal equivalence of analytic functions on Jordan domains to polynomial models, expanding understanding of function representation.

Findings

Any analytic function on a Jordan domain has a polynomial conformal model.

Existence of an injective analytic map transforming the domain for polynomial representation.

The proof offers a new geometric perspective on conformal equivalence.

Abstract

In this paper we present a geometric proof of the following fact. Let $D$ be a Jordan domain in $\mathbb{C}$, and let $f$ be analytic on $cl(D)$. Then there is an injective analytic map $\phi:D\to\mathbb{C}$, and a polynomial $p$, such that $f\equiv p\circ\phi$ on $D$ (that is, $f$ has a polynomial conformal model $p$).