On conformal maps from multiply connected domains onto lemniscatic domains (Q288623)

Let \(\hat{\mathbb C}\) denote the extended complex plane. Walsh's canonical domain is a \textit{lemniscatic domain} of the form NEWLINENEWLINE\[NEWLINE\mathcal L:=\big\{w\in\hat{\mathbb C}:|U(w)|>\mu\big\},\;\;U(w):=\prod_{j=1}^n(w-a_i)^{m_j},NEWLINE\]NEWLINE NEWLINEwhere \(a_j\in\mathbb C\), \(j=1,\dots,n\), are pairwise distinct, \(m_j>0\), \(j=1,\dots,n\), \(\sum_{j=1}^nm_j=1\), and \(\mu>0\). If \(\mathcal K\) is the exterior of \(n\geq1\) non-intersecting simply connected compact sets, then there exist a unique lemniscatic domain \(\mathcal L\) and a unique bijective conformal map \(\Phi:\mathcal K\to\mathcal L\) with \(\Phi(z)=z+O(\frac{1}{z})\) near infinity. The map \(\Phi\) is called a lemniscatic map. For a set \(\Omega\subset\mathbb C\), denote \(\Omega^*:=\{\overline z:z\in\Omega\}\). The authors derive a construction principle for the lemniscatic map under the conditions of the following theorem.NEWLINENEWLINETheorem 3.1: Let \(\Omega=\Omega^*\subseteq\mathbb C\) be compact and simply connected (not a single point) with exterior Riemann map \(\tilde\Phi:\hat{\mathbb C}\setminus\Omega\to\{w\in\hat{\mathbb C}:|w|>1\}\), \(\tilde{\Phi}(\infty)=\infty\), \(\tilde{\Phi}'(\infty)>0\). Let \(P(z)=\alpha z^n+\alpha_0\), \(\alpha>0\), \(n\geq2\), and \(\alpha_0\in\mathbb R\) to the left of \(\Omega\). Then \(E:=P^{-1}(\Omega)\) is the disjoint union of \(n\) simply connected compact sets, and NEWLINE\[NEWLINE\Phi:\hat{\mathbb C}\setminus E\to\mathcal L=\big\{w\in\hat{\mathbb C}:|U(w)|>\mu\big\},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\Phi(z)=z\left(\frac{\mu^n}{z^n}[(\tilde{\Phi}\circ P)(z)-(\tilde{\Phi}\circ P)(0)]\right)^{\frac{1}{n}},NEWLINE\]NEWLINE is the lemniscatic map of \(E\), where the principal branch of the \(n\)-th root is taken, and NEWLINE\[NEWLINE\mu:=\left(\frac{1}{\alpha\tilde{\Phi}'(\infty)}\right)^{\frac{1}{n}}>0, \;\;U(w):=(w^n+\mu^n(\tilde{\Phi}\circ P)(0))^{\frac{1}{n}}.NEWLINE\]NEWLINENEWLINENEWLINETheorem 3.1 is applied to obtain the lemniscatic conformal map for a radial slit domain. The authors also analytically construct the lemniscatic map of a set \(E\) that is the union of two disjoint equal disks.