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

We consider Walsh’s conformal map from the exterior of a set E=⋃j=1ℓEj consisting of 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] , 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 ℓ intervals. Once the centers are known, the corresponding Walsh map can be computed numerically. In addition, if E consists of ℓ= 2 or 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.