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On a non-compact [[Riemann surface]], every meromorphic function can be realized as a quotient of two (globally defined) holomorphic functions. In contrast, on a compact Riemann surface, every holomorphic function is constant, while there always exist non-constant meromorphic functions.
== Aporomorphy ==
In contrast to [[meromorphic functions]], which have only isolated poles, there is no universally established term in complex analysis for functions with essential singularities. While meromorphic functions are characterized by their “well-behaved” singularities, where the function diverges to infinity, functions with [[isolated singularity| isolated essential singularities]] exhibit far more complex behavior.
A function with isolated essential singularities could be called '''aporomorphic''' (from the Greek ''ἄπορος'' ''aporos, meaning “impassable” or “mysterious”''), although this term is not established in the mathematical literature. This designation would reflect the unpredictable and chaotic behavior of such functions near their singularities, as described by the Casorati–Weierstrass theorem.
== See also ==
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