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meromorphic function

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Idea

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

References

category: analysis

Revised on October 31, 2017 07:14:56 by Urs Schreiber (94.220.69.162)