Journal of the Mathematical Society of Japan

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

Buma L. FRIDMAN and Daowei MA

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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

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

First available in Project Euclid: 24 January 2013

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Zentralblatt MATH identifier

Primary: 30E99: None of the above, but in this section 30C99: None of the above, but in this section

analytic functions Hausdorff dimension Hartogs property


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.

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