The third Jacobi theta function
$$\theta_{3}\left(z,q\right)=1+2\sum_{n=1}^{\infty}q^{n^{2}}\cos\left(2\pi n z\right)$$
appears in the study of path integrals in QM. Specifically in the problem of a particle on a ring (see here). In that case it happens that $\left|q\right|=1$, rendering the function undefined. However, I am expecting a limit to exist for $\left|q\right|\uparrow1$ at least in the distribution sense. For instance (see here), if $q=1$
$$\theta_{3}\left(z,q\right)=\sum_{n=-\infty}^{\infty}\delta\left(z-n\right)$$
Can this result be generalized for any $q$ on the unit circle? Will we also get a comb of deltas? Maybe other well-known distributions? My goal is to get a more descriptive expression that can be visualized, if possible, including an understanding of what happens when I change $z,q$. Any directions/relevant references would be of help.
Thanks in advance!