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Holomorphic functions on subsets of \({\mathbb C}\). (English) Zbl 1270.30008

Summary: Let \(\Gamma\) be a \(C^\infty\) curve in \(\mathbb D\) containing \(0\); it becomes \(\Gamma_\theta\) after rotation by the angle \(\theta\) around \(0\). Suppose a \(C^\infty\) function \(f\) can be extended holomorphically to a neighborhood of each element of the family \(\{\Gamma_\theta\}\). We prove that under some conditions on \(\Gamma\) the function \(f\) is necessarily holomorphic in a neighborhood of the origin. In case \(\Gamma\) is a straight segment, the well known Bochnak-Siciak Theorem gives such a proof for real analyticity. We also provide several other results related to testing the holomorphy property on a family of certain subsets of a domain in \(\mathbb C\).

MSC:

30E99 Miscellaneous topics of analysis in the complex plane
30C99 Geometric function theory
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