10. Construction 2.13.1 - Weierstrauss P-functions

Let $L = ω_{1} Z + ω_{2} Z \subseteq C$ be a lattice. We define the weierstrauss $℘$ -function to be:

℘ (z) = \frac{1}{z^{2}} + \sum_{ω \in L - {0}} (\frac{1}{(z - ω)^{2}} - \frac{1}{ω^{2}})

Which is 1;;holomorphic and even.
Proof:

We use the following lemma:

L E M M A

The sum

\sum_{ω \in L - {0}} \frac{1}{| ω |^{λ}}

converges for $λ > 2$ .

Using this, we may show that the weierstrauss $℘$ -function is a local uniform limit of holomorphic functions, and is thus holomorphic itself.

One more observation