Continued fractions and conformal mappings for domains with angle points

TL;DR

This paper introduces a method for constructing conformal mappings using continued fractions, focusing on domains with angle points, and provides convergence analysis and illustrative examples.

Contribution

It presents a novel approach to conformal mapping construction via continued fractions, including convergence estimates and practical examples.

Findings

Convergence of continued fraction approximations to algebraic roots is established.

Explicit estimates for convergence rate and domain are provided.

Practical examples demonstrate the effectiveness of the method.

Abstract

Here we construct the conformal mappings with the help of continuous fractions approximations. These approximations converge to the algebraic roots $\sqrt[N]{z}$ for $N \in \mathbb{N}$ and $z$ from the right half-plane of the complex plane. We estimate both the convergence rate and the compact set of convergence. Also we give the examples that illustrate the introduced technique of a conformal mapping construction.