Walsh's Conformal Map onto Lemniscatic Domains for Several Intervals

We consider Walsh’s conformal map from the complement of a compact set $E={\cup }_{j=1}^{\mathrm{\ell }}{E}_j$ with $\mathrm{\ell }$ components onto a lemniscatic domain $\widehat{\mathbb{C}}\setminus L$, where $L$ has the form $L=\{w\in \mathbb{C}:{\prod }_{j=1}^{\mathrm{\ell }}\mid w-a_j{\mid }^{m_j}\le \mathrm{cap}(E)\}$. We prove that the exponents $m_j$ appearing in $L$ satisfy $m_j={\mu }_E(E_j)$, where ${\mu }_E$ is the equilibrium measure of $E$. When $E$ is the union of $\mathrm{\ell }$ real intervals, we derive a fast algorithm for computing the centers $a_1,\dots ,a_{\mathrm{\ell }}$. For $\mathrm{\ell }=2$, the formulas for $m_1,m_2$ and $a_1,a_2$ are explicit. Moreover, we obtain the conformal map numerically. Our approach relies on the real and complex Green’s functions of $\widehat{\mathbb{C}}\setminus E$ and $\widehat{\mathbb{C}}\setminus L$.