Walsh’s Conformal Map onto Lemniscatic Domains for Polynomial Pre-images I

Abstract We consider Walsh’s conformal map from the exterior of a compact set $$E \subseteq \mathbb {C}$$ E ⊆ C onto lemniscatic domain. If E is simply connected, domain circle, while if has several components, generalized lemniscate and determined by logarithmic capacity exponents centers lemniscate. For general , we characterize in terms Green’s function $$E^c$$ c . Under additional symmetry conditions on also locate polynomial pre-images = P^{-1}(\Omega )$$ = P - 1 ( Ω ) simply-connected infinite $$\Omega $$ explicitly determine derive equations to Finally, present examples where obtain domain, as well map.