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

Klaus Schiefermayr, Olivier Sète

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


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 languageEnglish
Publication statusAccepted/In press - 2023


  • Critical values
  • Green’s function
  • Lemniscatic domain
  • Logarithmic capacity
  • Multiply connected domain
  • Polynomial pre-image
  • Walsh’s conformal map


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