meromorphic functions of several variables

Definition.

Let $\Omega\subset{\mathbb{C}}^{n}$ be a domain and let $h\colon\Omega\to{\mathbb{C}}$ be a function. $h$ is called if for each $p\in\Omega$ there exists a neighbourhood $U\subset\Omega$ ($p\in U$) 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 $\infty$ 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 ${\mathbb{C}}^{2}$ 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 MeromorphicFunctionsOfSeveralVariables 2013-03-22 16:01:10 2013-03-22 16:01:10 jirka (4157) jirka (4157) 4 jirka (4157) Definition msc 32A20 indeterminancy set