Difference between revisions of "Aufgaben:Problem 1"

Revision as of 14:07, 31 December 2014

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. \(\square\)

Part b)

\( \vdash: \) Let \( Q = \left[ 0, 1\right] \times \left[ 0, 1 \right] \subset \mathbb{C} \) be the unit square, and let \( f \) be a holomorphic function on a neighborhood of \( Q \). Suppose further that \( f( z + i) - f(z) \in \mathbb{R}_{\geq 0} \) for any \( z \in \left[ 0, 1 \right] \) and \( f( z + 1) - f(z) \in \mathbb{R}_{\geq 0} \) for any \( z \in \left[ 0, i \right] \). Then \(f\) is constant.

Proof

We write \( z = x + iy \) and \( f(x + iy) = u(x,y) + i v(x,y) \) where \( u, v \) are real valued functions. By assumption we get:

$$ \forall y \in \left[ 0, 1 \right] : v(1, y) = v(0, y) $$

$$ \forall x \in \left[ 0, 1 \right] : v(x, 0) = v(x, 1) $$

We now prove a little Lemma:

Pierre's Lemma: If \( f = u + iv \) is a holomorphic function then \( u, v \) are meromorphic.

\( \color{red}{So \: beautiful! \: See \: discussion} \)

Proof of Lemma:

If \( a, b \in \mathbb{R} \smallsetminus \{ 0 \} \), then \( a \) and \( b i \) are linearly independent. ( \( \mathbb{C} \) is a two-dimensional \( \mathbb{R} \)-vectorspace with basis \( 1 \) and \( i \). ) If \( v \) is zero then \( f = u \) is constant (see critic in discussion), if \( u = 0 \) then \( v \) is meromorphic anyway and yay. \( \square \)

We thus see that \( v \) can be doubly-periodic, analytically continued on the lattice \( \Omega = \{ m + in : m, n \in \mathbb{Z} \} \). So we make \( v \) an elliptic function.

We've shown in part a) that a elliptic, holomorphic function has to be constant. Thus \( v \) is constant.

From the Cauchy-Riemann equations it follows immediately that also \( u \) has to be constant. \( \square \)