Does anybody know this relation between Jacobi theta functions?

Question

I'm proving a theorem that I know is correct, but I would need to proof that $$ \theta{}_{2}\left(z\right)\theta{}_{3}\left(z\right)=\theta'{}_{4}\left(z\right)\theta{}_{1}\left(z\right)-\theta'{}_{1}\left(z\right)\theta{}_{4}\left(z\right) $$ Where $\theta{}_{k}\left(z\right)$ are the Jacobi theta functions. I know that is $z=0$ than you have the very well known Riemann identity, but here I'm interested in generic $z$. Does anybody have seen it before? O does anybody have an idea on how to prove it?

Answer 1

Equation (2.7.2) on p. 57 in Pi and the AGM by Borwein & Borwein reads $$ \frac{d}{dz} \frac{\theta_1(z)}{\theta_4(z)} = \theta_4(0)^2 \, \frac{\theta_2(z) \theta_3(z)}{\theta_4(z)^2} , $$ which almost agrees with your formula; you seem to be missing a factor $-\theta_4(0)^2$.

They outline a proof in an exercise on p. 59. (Roughly: show that $(\theta_1' \theta_4 - \theta_4' \theta_1)/\theta_2 \theta_3$ is doubly periodic and has poles with certain properties, and hence is constant.)