I have a complex plane with two horizontal cuts $[-\alpha \pm i/2, \alpha \pm i/2]$ for real $\alpha$. We can imagine gluing the two cuts to get a torus with complex parameter $\tau$. Thinking of the fundamental parallelogram of the torus, we can have the "upper-most" A-cycle of the torus map into the cut on the complex plane.
According the the literature, the function $$-\frac{1}{2} \frac{\theta'_{3}(\pi x | \tau)}{\theta_{3}(\pi x | \tau)}, \,\,\,\,\,\,\,\, x \in \big[-\frac{1}{2}, \frac{1}{2}\big]$$
maps the upper-most A-cycle of the torus to the subset $[-\alpha , \alpha]$ of the real axis. (The theta functions are of course the Jacobi Theta Functions.) However, when I try to verify this using Mathematica, I find that only when $\rm{Re}(\tau)=0$ does that function map nicely into the real numbers; otherwise, it has imaginary parts too.
Any idea why this is happening? I need that function to map into the real numbers for all $\tau \in \mathbb{H} \, / \, \rm{SL}(2, \mathbb{Z}) $.