Meromorphic Function is Elliptic iff Doubly Periodic

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

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Historical Note

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

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.33$: Weierstrass ($\text {1815}$ – $\text {1897}$)