I am currently working through Landau's mechanics book and I am struggling to get my head around a solution provided in Chapter.6 relating to the asymmetrical top.
I fully comprehend how Landau parametrises the motion in absolute space with Eulerian angles and have managed to derive appropriate solutions for angles $\theta(t)$ and $\psi(t)$. I also managed to obtain the correct expression for $$\frac{d\phi(t)}{dt} = \frac{A\text{sn($\alpha$ t,k)}^2 + B\text{cn($\alpha$ t,k)}^2}{C\text{sn($\alpha$ t,k)}^2 + D\text{cn($\alpha$ t,k)}^2}$$ for appropriate constants A,B,C,D,$\alpha$. He refers us to Whittaker, Analytical Dynamics Chap.6 for the solution derivation. From my understanding Whittaker first finds critical points of the solution (nodes and zeroes), and then associates a Jacobi-theta function to the solution as these $\theta_{ab}$ can be defined by their critical points. (I find Whittaker's working of the problem a bit "grey" at times but I think I get the main idea).
Now Landau provides a solution in terms of $\theta_{10}$ functions: $$\phi(t) = \frac{1}{2i}\ln[\frac{\theta_{01}(2t/T - i\alpha)}{\theta_{01}(2t/T + i\alpha)}] + Xt$$
for X another combination of theta-functions. Now $\theta_{10}$ is defined as a series expansion: $$\theta_{10}(z,\tau) = \sum_{n=-\infty}^{\infty} \text{exp}[i\pi\tau n^2+2i\pi n(z+\frac{1}{2})]$$
I believe my misunderstanding comes from the notation of Landau. By definition we must have that $\tau \in \mathbb{C}$ Upper-Half plane and $z \in \mathbb{C}$. Could anyone clarify the single variable notation used by Landau? It is impossible to set $\tau = 0$ as it must lie in the Upper-Half plane, and if we set $z = 0$ then I am not sure of the physicality of this solution (even though it is an angle and would be taken as modulo-$2\pi$ ?), I can't manage to get a picture of the function's behaviour in this case...
Any help would be very much appreciated !