TY - JOUR
T1 - Walsh’s Conformal Map Onto Lemniscatic Domains for Polynomial Pre-images II
AU - Schiefermayr, Klaus
AU - Sète, Olivier
N1 - Publisher Copyright:
© The Author(s) 2023.
PY - 2024/6
Y1 - 2024/6
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
KW - 30C20
KW - 65E10
KW - 30C35
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
VL - 24
SP - 257
EP - 281
JO - COMPUTATIONAL METHODS AND FUNCTION THEORY
JF - COMPUTATIONAL METHODS AND FUNCTION THEORY
IS - 2
ER -