$$\wp(z;\tau) = -(\log \vartheta_{11}(z;\tau))'' + c $$ $$\vartheta_{11}(z|q) = -2 q^{1/4}\sin(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m}+q^{4m}\right)$$
Then we will get that $\wp$ is sum over m. I can write it but it is a bit disaster.
Questions 1: Why have not I seen this formula before? Why there is no such formula in wiki?
As far as I understand it has to converge cause original formula for theta function converge uniformly on compacts (in complex analysis uniformly convergence of series implies convergence of their derivatives).
Questions 2: If it is really make sence then what is relation of this formular to classical one (double sum)?