# Riemann’s removable singularity theorem in several variables

###### Theorem.

Suppose $V$ is a proper analytic variety in an open set $U\mathrm{\subset}{\mathrm{C}}^{n}$ (that is of dimension^{} at most $n\mathrm{-}\mathrm{1}$)
suppose that $f\mathrm{:}U\mathrm{\backslash}V\mathrm{\to}\mathrm{C}$ is holomorphic and further that $f$ is locally bounded in $U$
Then there exists a unique holomorphic
extention of $f$ to all of $U$.

If $V$ is of even lower dimension we can in fact even drop the locally bounded requirement, see the Hartogs extension theorem.

## References

- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
- 2 Hassler Whitney. . Addison-Wesley, Philippines, 1972.

Title | Riemann’s removable singularity^{} theorem in several variables |
---|---|

Canonical name | RiemannsRemovableSingularityTheoremInSeveralVariables |

Date of creation | 2013-03-22 15:34:57 |

Last modified on | 2013-03-22 15:34:57 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 4 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 30D30 |

Classification | msc 32H02 |

Synonym | Riemann’s extension theorem |