Holomorphic functions on subsets of C

Abstract

Let $\Gamma$ be a $C^{\infty }$ curve in $\mathbb{C}$ containing 0; it becomes ${\Gamma }_{\theta }$ after rotation by angle $\theta$ about 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 \textit{real analyticity}. We also provide several other results related to testing holomorphy property on a family of certain subsets of a domain in $\mathbb{C}$.

12 pages

Cited by in corpus (1)

• Testing holomorphy on curves