Meromorphic Function is Elliptic iff Doubly Periodic

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Theorem

Let $f: \C \to \C$ be a meromorphic function.

Then $\map f z$ is an elliptic function if and only if it is also doubly periodic.


Proof


Historical Note

Meromorphic Function is Elliptic iff Doubly Periodic was discovered by Niels Henrik Abel‎ during his exploration of elliptic functions.


Sources