Multivalued Analytic Continuation of the Cauchy Transform |
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Authors: | Lavi Karp |
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Institution: | (1) Department of Mathematics, ORT Braude College, PO Box 78, 21982 Karmiel, Israel |
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Abstract: | In this paper we introduce the notion of multivalued analytic continuation of the Cauchy transforms. Many difficulties arise
because the continuation is not single-valued. Our main result asserts that if χΩ has a multivalued analytic continuation, then the free boundary ∂Ω has zero Lebesgue measure. Here χΩ is the characteristic function of a domain Ω and ∂Ω is its boundary. We also discuss the connections between this notion,
quadrature domains and approximations of analytic functions with single-valued integrals by rational functions. The last problem
is related to the existence of a continuous function g and a closed connected set K such that the gradient of g vanishes on K, nevertheless g is not constant on K.
Mathematics Subject Classifications (2000) Primary 31A25, 31B20; secondary 30E10, 35J05, 41A20. |
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Keywords: | Cauchy transform multivalued analytic continuation quadrature domains |
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