In his proof of the $\eta$- transformation formula Elsrodt uses the Weierstrass $\zeta$ function. His first step in the proof is the following:
Let $\mathbb{H}$ denote the upper half-plane in $\mathbb{C}$ and consider the Weierstrass $\zeta$ function.
$\begin{equation} \zeta(z)=\frac{1}{z}+\sum_{m,n\in\mathbb{Z}}'(\frac{1}{z+m\tau+n}-\frac{1}{m\tau+n}+\frac{z}{(m\tau+n)^2}) \end{equation}$
for the lattice $\Lambda=\mathbb{Z}+\mathbb{Z}\tau$ where $\tau\in\mathbb{C}/\Lambda$ and where the prime in the sum means that $(m,n)=(0,0)$ is excluded from the summation. Splitting off the terms with $m=0$ in the above equation and using Euler's partial fraction expansion of $\pi\cot\pi u$ and $\pi^2/\sin^2\pi u$ we obtain
$\zeta(z)=\frac{\pi^2}{3}z+\pi\cot\pi z+\sum_{m\neq0}(\pi\cot\pi(m\tau+z)-\pi\cot\pi m\tau+\frac{z\pi^2}{\sin^2\pi m\tau})$
I'm assuming he is referring to the following series representations of the trigonometric functions: $\frac{\pi}{\sin\pi u}=\frac{1}{u}+\sum_{n=1}^{\infty}\frac{2u}{n^2-u^2}$
and
$\pi\cot\pi u=\frac1u+\sum_{n=1}^\infty(\frac1{u-n}+\frac1{u+n})$
I've been trying for a while now to understand the transformation, but I'm coming up empty. Am I missing something that makes this obvious or is it a purely computational thing and I'm making a mistake trying to find the equality computationally? Any help would be greatly appreciated.