Let $\Lambda$ be a lattice and $\sigma_{\Lambda_{\tau}}(z):= \sigma(z)= \prod_{w\in \Lambda\setminus \{0\}} \left(1 - \frac{z}{w}\right)\exp\left(\frac{z}{w}+\frac{z^2}{2w^2}\right) $ the Weierstraß sigma function. Furthermore let $\eta_1$ be a quasiperiod of the Weierstraß $\zeta$ function.
For $q = \exp(2 \pi i \tau)$ and $u = \exp(2 \pi i z)$ the following identity holds \begin{align} \sigma_{\Lambda_{\tau}}(z) = \frac{1}{2 \pi i} \exp\left(\frac{\eta_1 z^2}{2}\right)\left(u^{\frac{1}{2}}- u^{-\frac{1}{2}}\right)\prod_{n=1}^{\infty} \frac{(1- q^nu)(1 -q^n u^{-1})}{(1-q^n)^2}. \end{align} I dont really know on how to prove it so i would be glad if anyone could help here. Thanks in advance.