Let $\vartheta_{00}$ and $\vartheta_{01}$ be Jacobian theta functions (notations like on wikipedia). $$F:=\left\{ \tau \in \mathbb{C}: \mathrm{Im}(\tau)>0, \left| \mathrm{Re}(\tau)\right|<1, \left|\mathrm{Re}\left(\frac{1}{\tau}\right)\right|\leq 1 \right\}.$$
I need to prove that $$ k'(\tau):=\left(\frac{\vartheta_{01}(0,\tau)}{\vartheta_{00}(0,\tau)}\right)^2 $$ is injective on $F$.
I tried to do so by looking at the theta functions' product expansions but did not succeed. Their power series did not help me either, but I am probably missing something!
How do you normally prove injectivity of a power series or an infinte product?
Suggestions are very appreciated!
write k' = N^2/D^2 you want to show that f(t) = N(t)^2 - a D(t)^2 has one zero for any a. Find equations relating f(t) on pairs of edges and integrate d(log f(t))/(2 pi i) around the edges of F.