Holomorphic functions on subsets of C

Buma L. Fridman; Daowei Ma

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arXiv · 1006.3105

Holomorphic functions on subsets of C

Buma L. Fridman·Daowei Ma

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Abstract

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

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BibTeX RIS

Buma L. Fridman, Daowei Ma. 2011-02-28. Holomorphic functions on subsets of C. https://arxiv.org/abs/1006.3105

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