---
tag: W6FTSF57T1
kind: concept
name: Meromorphic function of several variables
msc: [32A20]
sources:
  - url: https://en.wikipedia.org/wiki/Meromorphic_function
---

## Summary

A meromorphic function of several variables is locally a quotient of two [[holomorphic functions|507C34W0ZT]] on an open set of $\C^n$.
The denominator must not be identically zero.
It extends the one-variable notion of a [[meromorphic function|0ZZ7ZYDR8W]].
A new feature for $n \ge 2$ is the presence of points of indeterminacy, where the function has no value at all, not even $\infty$.
Meromorphic functions are the natural analogue of rational functions on complex manifolds and analytic spaces.

## Explanations

### L1 human

Rational functions such as $z_1 / z_2$ are the simplest meromorphic functions on $\C^2$.
This function has poles along the line $z_2 = 0$, except at the origin.
Near the origin it takes every complex value: along the line $z_1 = c z_2$ it is constantly equal to $c$.
So at the origin it cannot be given any value, finite or infinite; the origin is a point of indeterminacy.
In one variable this never happens, because poles are isolated points where the function tends to $\infty$.