nLab meromorphic function

Idea

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).

Related concepts

• entire function

• sheaf of meromorphic functions

• function field of a variety

References

• Wikipedia, Meromorphic function

• eom, Meromorphic function

category: analysis