# meromorphic functions of several variables

###### Definition.

Let $\mathrm{\Omega}\subset {\u2102}^{n}$ be a domain and let $h:\mathrm{\Omega}\to \u2102$ be a function. $h$ is called if for each $p\in \mathrm{\Omega}$ there exists a neighbourhood $U\subset \mathrm{\Omega}$ ($p\in U$) and two holomorphic (http://planetmath.org/HolomorphicFunctionsOfSeveralVariables) functions $f,g$ defined in $U$ where $g$ is not identically zero, such that $h=f/g$ outside the set where $g=0$.

Note that $h$ is really defined only outside of a complex analytic subvariety. Unlike in one variable, we cannot simply define $h$ to be equal to $\mathrm{\infty}$ at the poles and expect $h$ to be a continuous mapping to some larger space (the Riemann sphere in the case of one variable). The simplest counterexample^{} in ${\u2102}^{2}$ is $(z,w)\mapsto z/w$, which does not have a unique limit at the origin. The set of points where there is no unique limit, is called the indeterminancy set. That is, the set of points where if $h=f/g$, and $f$ and $g$ have no common factors, then the indeterminancy set of $h$ is the set where $f=g=0$.

## References

- 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | meromorphic functions of several variables |
---|---|

Canonical name | MeromorphicFunctionsOfSeveralVariables |

Date of creation | 2013-03-22 16:01:10 |

Last modified on | 2013-03-22 16:01:10 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 4 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32A20 |

Defines | indeterminancy set |