TY - JOUR

T1 - Walsh’s Conformal Map Onto Lemniscatic Domains for Polynomial Pre-images II

AU - Schiefermayr, Klaus

AU - Sète, Olivier

N1 - Publisher Copyright:
© 2023, The Author(s).

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

KW - Critical values

KW - Green’s function

KW - Lemniscatic domain

KW - Logarithmic capacity

KW - Multiply connected domain

KW - Polynomial pre-image

KW - Walsh’s conformal map

UR - http://www.scopus.com/inward/record.url?scp=85166919236&partnerID=8YFLogxK

U2 - 10.1007/s40315-023-00492-6

DO - 10.1007/s40315-023-00492-6

M3 - Article

AN - SCOPUS:85166919236

SN - 1617-9447

JO - COMPUTATIONAL METHODS AND FUNCTION THEORY

JF - COMPUTATIONAL METHODS AND FUNCTION THEORY

ER -