## Holomorphic functions on subsets of $${\mathbb C}$$.(English)Zbl 1270.30008

Summary: Let $$\Gamma$$ be a $$C^\infty$$ curve in $$\mathbb D$$ containing $$0$$; it becomes $$\Gamma_\theta$$ after rotation by the angle $$\theta$$ around $$0$$. 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 the holomorphy property on a family of certain subsets of a domain in $$\mathbb C$$.

### MSC:

 30E99 Miscellaneous topics of analysis in the complex plane 30C99 Geometric function theory

### Keywords:

analytic functions; Hausdorff dimension; Hartogs property
Full Text:

### References:

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