Aufgaben:Problem 1

Part a)

Prove that, if \(f\) is an elliptic function, then \(f\) is constant if and only if \(f\) has no poles.

Solution part a)

Definition (elliptic function): A function \(f\) is called elliptic if it has the following two properties:

(a) \(f\) is doubly periodic.

(b) \(f\) is meromorphic (its only singularities in the finite plane are poles).

(Taken from Apostol, 1.4)

Liouville's theorem: If \(f\) is holomorph on \( \mathbb{C} \) and bounded, then \(f\) is constant.

To prove: Let \(f\) be an elliptic function: \(f\) is constant \(\Leftrightarrow\) \(f\) has no poles.

\( "\Rightarrow" \) Let \(f\) be constant on \( \mathbb{C} \) \( \rightarrow f\) has no poles.

\( "\Leftarrow" \) Let \(P\) be the fundamental region to \(f\), where \(f\) is elliptic and has no poles.

\( f(\mathbb{C}) = f(P) \) is compact and thus bounded. Because \(f\) is both holomorph and bounded on \(\mathbb{C}\), Liouville's theorem tells us that \(f\) has to be a constant function. \(\square\)