Holomorphic functions on subsets of ℂ

概要

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Let Γ be a C∞ curve in ℂ containing 0; it becomes Γθ after rotation by angle θ about 0. Suppose a C∞ function f can be extended holomorphically to a neighborhood of each element of the family {Γθ}. 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 real analyticity. We also provide several other results related to testing holomorphy property on a family of certain subsets of a domain in ℂ.
Mathematical Society of Japanの論文