Behaviour of elliptic functions near degenerate lattice

Question

What can be said about elliptic functions (Weierstrass $\wp$, $\sigma$, Jacobi $\theta$, sn, cn, etc.) in the limit of degenerate lattice. By "degenerate" I mean $\tau = \omega_3/\omega_1$ tends to a non-zero, finite real number or $q = \exp(i\pi\tau)$ tends to a point on the unit circle but not to 1.

In the literature I find many results for $q \to 0$ or $q \to 1$, mostly with rectangular lattice ($\tau$ purely imaginary). So I specifically excluded these cases from this question.

I'm aware that the behaviour of a single function is quite crazy for general degenerate lattices. I'm actually hoping for a miracle for non-trivial combinations of elliptic functions, something like $\sqrt{\theta_1/\theta_4}$.

Answer 1

When one period is fixed while another tends to zero, elliptic functions degenerate to elementary functions (combinations of trigonometric and rational). It does not matter whether the lattice is rectangular or not. When both periods tend to infinity, they degenerate to rational functions. For explicit formulas see, for example the book of Akhiezer, Elements of the theory of elliptic functions, table VII.