\[\theta_{4}\left(z\middle|\tau\right)=\theta_{4}\left(0\middle|\tau\right)\*%
\lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty}\prod_{m=-M}^{M}\left(1+%
\frac{z}{(m+(n-\tfrac{1}{2})\tau)\pi}\right).\]