Multivalued analytic continuation of the Cauchy transform
type: Item
instance of: scholarly article
title: Multivalued analytic continuation of the Cauchy transform
zbMATH Open document ID: 1090.31001
author: Lavi Karp
published in: Potential Analysis
publication date: 2006-03-21T00:00:00Z
review text: The Cauchy transform \[ {\widehat{g}}(z) = -\frac{1}{\pi} \int \frac{g(w)}{w - z} \,dA \] well-known in potential theory and applications, is studied. The following notion of the multivalued analytic continuation of the Cauchy transform of a characteristic function \(\chi_{\Omega}\) of a bounded domain \(\Omega\) is introduced: the Cauchy transform \({\widehat{\chi}}_{\Omega}\) possesses a multivalued analytic continuation in a neighborhood of \(z_0\in \partial \Omega\) if for certain positive \(r\) there exists an analytic function \(f(z)\) in \(B_r(z_0)\) such that for any (connected) component \(K\) of \(B_r(z_0)\setminus \Omega\) \[ {\widehat{\chi}}_{\Omega} = f(z) + c_K,\quad z\in K, \] where \(c_K\) is a constant depending on \(K\). The author proves the following property of such a continuation: if \({\widehat{\chi}}_{\Omega}\) has a multivalued continuation in \(B_r(z_0)\), \(z_0\in \partial \Omega\), then (a) \(\partial \Omega \cap {B}_r(z_0)\) has Lebesgue measure zero; (b) for any \(a\in \partial \Omega \cap {B}_r(z_0)\) \[ \int_{B_{\rho}(a)} \frac{\chi_{\Omega}(z)}{| z - a| ^2} \,dA = \infty \] for certain \(\rho > 0\). Connection of the introduced notion with the quadrature domains and approximation by rational functions are discussed too. An open problem is formulated.
Mathematics Subject Classification ID: 31A25
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 76D27
Mathematics Subject Classification ID: 76M40
Mathematics Subject Classification ID: 30B40
zbMATH DE Number: 5014106
zbMATH Keywords: Cauchy transform
zbMATH Keywords: multivalued analytic continuation
zbMATH Keywords: quadrature domains
zbMATH Keywords: approximation by rational functions
reviewed by: Sergei V. Rogosin
MaRDI profile type: Publication
full work available at URL: s11118-005-8566-z
OpenAlex ID: W1968508501
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DOI: 10.1007/S11118-005-8566-Z
description: scientific article; zbMATH DE number 5014106