Walsh's Conformal Map onto Lemniscatic Domains for Polynomial Pre-images II

TL;DR

This paper develops an iterative method to compute the centers of Walsh's conformal map for polynomial preimages of intervals, enabling numerical computation of the map, especially for symmetric cases with explicit formulas.

Contribution

It introduces a new iterative approach for determining centers of Walsh's conformal maps for polynomial preimages, extending previous formulas and providing explicit solutions for symmetric cases.

Findings

Iterative method for computing centers of Walsh maps.

Explicit formulas for symmetric cases with two or three components.

Numerical examples demonstrating the method's effectiveness.

Abstract

We consider Walsh's conformal map from the exterior of a set $E=\bigcup_{j=1}^\ell E_j$ consisting of $\ell$ compact disjoint components onto a lemniscatic domain. In particular, we are interested in the case when $E$ is a polynomial preimage of $[-1,1]$, i.e., when $E=P^{-1}([-1,1])$, where $P$ is an algebraic polynomial of degree $n$. Of special interest are the exponents and the centers of the lemniscatic domain. In the first part of this series of papers, a very simple formula for the exponents has been derived. In this paper, based on general results of the first part, we give an iterative method for computing the centers when $E$ is the union of $\ell$ intervals. Once the centers are known, the corresponding Walsh map can be computed numerically. In addition, if $E$ consists of $\ell=2$ or $\ell=3$ components satisfying certain symmetry relations then the centers and the corresponding Walsh map are given by explicit formulas. All our theorems are illustrated with analytical or numerical examples.