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meromorphic functions of several variables
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(Definition)
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Definition 1 Let $\Omega \subset {\mathbb{C}}^n$ be a domain and let $h \colon \Omega \to {\mathbb{C}}$ be a function. $h$ is called meromorphic if for each $p \in \Omega$ there exists a neighbourhood $U \subset \Omega$ ( $p \in U$ and two holomorphic 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 $f = g = 0$
- 1
- Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.
- 2
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"meromorphic functions of several variables" is owned by jirka.
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indeterminancy set |
This object's parent.
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Cross-references: factors, points, origin, limit, counterexample, Riemann sphere, continuous mapping, poles, variable, complex analytic subvariety, neighbourhood, function, domain
This is version 1 of meromorphic functions of several variables, born on 2006-06-19.
Object id is 8058, canonical name is MeromorphicFunctionsOfSeveralVariables.
Accessed 1621 times total.
Classification:
| AMS MSC: | 32A20 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Meromorphic functions) |
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Pending Errata and Addenda
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