## Abstract

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.

Original language | English |
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Pages (from-to) | 257-281 |

Number of pages | 25 |

Journal | COMPUTATIONAL METHODS AND FUNCTION THEORY |

Volume | 24 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 2024 |

## Keywords

- Critical values
- Green’s function
- Lemniscatic domain
- Logarithmic capacity
- Multiply connected domain
- Polynomial pre-image
- Walsh’s conformal map
- 30C20
- 65E10
- 30C35