## Journal of the Mathematical Society of Japan

### Holomorphic functions on subsets of ${\Bbb C}$

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

#### Article information

Source
J. Math. Soc. Japan, Volume 65, Number 1 (2013), 1-12.

Dates
First available in Project Euclid: 24 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1359036446

Digital Object Identifier
doi:10.2969/jmsj/06510001

Mathematical Reviews number (MathSciNet)
MR3034396

Zentralblatt MATH identifier
1270.30008

#### Citation

FRIDMAN, Buma L.; MA, Daowei. Holomorphic functions on subsets of ${\Bbb C}$. J. Math. Soc. Japan 65 (2013), no. 1, 1--12. doi:10.2969/jmsj/06510001. https://projecteuclid.org/euclid.jmsj/1359036446

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