Holomorphic functions on subsets of C

Abstract

Let $\Gamma $ be a $C^\infty $ curve in $\Bbb{C}$ 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 $\Bbb{C}$.