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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.07292 |
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| _version_ | 1866907792243163136 |
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| author | Schiefermayr, Klaus Sète, Olivier |
| author_facet | Schiefermayr, Klaus Sète, Olivier |
| contents | We consider Walsh's conformal map from the complement of a compact set $E = \cup_{j=1}^\ell E_j$ with $\ell$ components onto a lemniscatic domain $\widehat{\mathbb{C}} \setminus L$, where $L$ has the form $L = \{ w \in \mathbb{C} : \prod_{j=1}^\ell \lvert w - a_j \rvert^{m_j} \leq \operatorname{cap}(E) \}$. We prove that the exponents $m_j$ appearing in $L$ satisfy $m_j = μ_E(E_j)$, where $μ_E$ is the equilibrium measure of $E$. When $E$ is the union of $\ell$ real intervals, we derive a fast algorithm for computing the centers $a_1, \ldots, a_\ell$. For $\ell = 2$, 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 $\widehat{\mathbb{C}} \setminus L$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07292 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Walsh's Conformal Map onto Lemniscatic Domains for Several Intervals Schiefermayr, Klaus Sète, Olivier Complex Variables 30C35, 30C20, 30C85 We consider Walsh's conformal map from the complement of a compact set $E = \cup_{j=1}^\ell E_j$ with $\ell$ components onto a lemniscatic domain $\widehat{\mathbb{C}} \setminus L$, where $L$ has the form $L = \{ w \in \mathbb{C} : \prod_{j=1}^\ell \lvert w - a_j \rvert^{m_j} \leq \operatorname{cap}(E) \}$. We prove that the exponents $m_j$ appearing in $L$ satisfy $m_j = μ_E(E_j)$, where $μ_E$ is the equilibrium measure of $E$. When $E$ is the union of $\ell$ real intervals, we derive a fast algorithm for computing the centers $a_1, \ldots, a_\ell$. For $\ell = 2$, 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 $\widehat{\mathbb{C}} \setminus L$. |
| title | Walsh's Conformal Map onto Lemniscatic Domains for Several Intervals |
| topic | Complex Variables 30C35, 30C20, 30C85 |
| url | https://arxiv.org/abs/2402.07292 |