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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 real value 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 $\sqrt{\theta_1/\theta_4}$.

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 $\sqrt{\theta_1/\theta_4}$.

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 real value 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). 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 $\sqrt{\theta_1/\theta_4}$.

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 real value 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 $\sqrt{\theta_1/\theta_4}$.

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 real value 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 $q \to 1$ with rectangular lattice ($\tau$ purely imaginary). 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 $\sqrt{\theta_1/\theta_4}$.

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 real value 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). 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 $\sqrt{\theta_1/\theta_4}$.