Walsh’s Conformal Map onto Lemniscatic Domains for Several Intervals

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Abstract We consider Walsh’s conformal map from the complement of a compact set $$E = \cup _{j=1}^\ell E_j$$ E = ∪ j = 1 ℓ E j with $$\ell $$ ℓ components onto a lemniscatic domain $$\widehat{\mathbb {C}} \setminus L$$ C ^ \ L , where L has the form $$L = \{ w \in \mathbb {C}: \prod _{j=1}^\ell |w - a_j|^{m_j} \le {{\,\textrm{cap}\,}}(E) \}$$ L = { w ∈ C : ∏ j = 1 ℓ | w - a j | m j ≤ cap ( E ) } . We prove that the exponents $$m_j$$ m j appearing in L satisfy $$m_j = \mu _E(E_j)$$ m j = μ E ( E j ) , where $$\mu _E$$ μ E is the equilibrium measure of E. When E is the union of $$\ell $$ ℓ real intervals, we derive a fast algorithm for computing the centers $$a_1, \ldots , a_\ell $$ a 1 , … , a ℓ . For $$\ell = 2$$ ℓ = 2 , the formulas for $$m_1, m_2$$ m 1 , m 2 and $$a_1, a_2$$ 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 $$\widehat{\mathbb {C}} \setminus E$$ C ^ \ E and $$\widehat{\mathbb {C}} \setminus L$$ C ^ \ L .