nLab meromorphic function

complex geometry

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analytic geometry (complex, rigid, global)

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analytic function

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Let $\Omega$ be a domain in a complex manifold and let $P \subset \Omega$ be a (complex-) analytic subset which is empty or of codimension one. A holomorphic function $f$ defined on the complement $\Omega \setminus P$ is called a meromorphic function in $\Omega$ if for every point $p \in P$ one can find an arbitrarily small neighbourhood $U$ of $p$ in $\Omega$ and functions $\phi$, $\psi$ holomorphic in $U$ without common non-invertible factors in $Int(U)$, such that $f = \phi/\psi$ in $U \setminus P$.

In one complex dimension (one complex variable), hence on a Riemann surface, a meromorphic function is a complex-analytic function which is defined away from a set of isolated points. Equivalently this is a holomorphic function with values in the Riemann sphere. Compare a holomorphic function, which is valued in the complex plane (the Riemann sphere minus a point).

category: analysis