### Abstract

The main result of the paper is the following generalization of Forelli's theorem (Math. Scand. 41:358-364, 1977): Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with eigenvalues whose ratios are positive reals. Then any function θ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary.

Original language | English (US) |
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Pages (from-to) | 655-666 |

Number of pages | 12 |

Journal | Journal of Geometric Analysis |

Volume | 19 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2009 |

### Keywords

- Holomorphic functions
- Holomorphic vector fields

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

Kim, K. T., Poletsky, E., & Schmalz, G. (2009). Functions holomorphic along holomorphic vector fields.

*Journal of Geometric Analysis*,*19*(3), 655-666. https://doi.org/10.1007/s12220-009-9078-7