Walsh’s Conformal Map Onto Lemniscatic Domains for Polynomial Pre-images II

Abstract

Abstract We consider Walsh’s conformal map from the exterior of a set $$E=\bigcup _{j=1}^{\ell }E_j$$ E = ⋃ j = 1 ℓ E j consisting of $$\ell $$ ℓ compact disjoint components onto a lemniscatic domain. In particular, we are interested in the case when E is a polynomial preimage of $$[-1,1]$$ [ - 1 , 1 ] , i.e., when $$E=P^{-1}([-1,1])$$ 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 $$\ell $$ ℓ intervals. Once the centers are known, the corresponding Walsh map can be computed numerically. In addition, if E consists of $$\ell =2$$ ℓ = 2 or $$\ell =3$$ ℓ = 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.