Let $f:\mathbb{C}\to \mathbb{C}$ be elliptic with lattice $L={\omega }_1\mathbb{Z}+{\omega }_2\mathbb{Z}$. Then we have the following:

1. $f(z)=f_1(z)+f_2(z){\mathrm{\wp }}^{\prime }(z)$ for $f_1,f_2$ 1;;even elliptic functions.
2. If $f$ is even, then it is a rational function in 1;;$\mathrm{\wp }(z)$

PROOF

1. Let $f_1(z)=\frac{f(z)+f(-z)}{2}$ and $f_2(z)=\frac{f(z)-f(-z)}{2\mathrm{\wp }(z)}$
2. If $a_1,\dots ,a_n$ are the zeros of $f$, and $b_1,\dots ,b_n$ the poles, then

is constant (show it has no zeros and no poles).