Open Access
January, 2013 Holomorphic functions on subsets of ${\Bbb C}$
Buma L. FRIDMAN, Daowei MA
J. Math. Soc. Japan 65(1): 1-12 (January, 2013). DOI: 10.2969/jmsj/06510001


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 real analyticity. We also provide several other results related to testing holomorphy property on a family of certain subsets of a domain in $\Bbb{C}$.


Download Citation

Buma L. FRIDMAN. Daowei MA. "Holomorphic functions on subsets of ${\Bbb C}$." J. Math. Soc. Japan 65 (1) 1 - 12, January, 2013.


Published: January, 2013
First available in Project Euclid: 24 January 2013

zbMATH: 1270.30008
MathSciNet: MR3034396
Digital Object Identifier: 10.2969/jmsj/06510001

Primary: 30C99 , 30E99

Keywords: analytic functions , Hartogs property , Hausdorff dimension

Rights: Copyright © 2013 Mathematical Society of Japan

Vol.65 • No. 1 • January, 2013
Back to Top