Define the function $w(x):= \frac{\theta_2(x)}{\theta_1(x)}$, where $x$ is a point in Jacobian variety of torus $T^2$, namely $x$ in the argument means $\int_{P_0}^x \omega$, $\omega$ being the holomorphic differential and $P_0$ some base point, and I left implicit the dependence on period matrix $\tau$ of $T^2$; $\theta_1$ the odd Riemann theta function for genus $g=1$, while $\theta_2$ is the one with characteristic $(1/2,0)$.
I want to compute the integral $$I:= \lim_{\epsilon \to 0} \int_{T^2} d^2x \int_{T^2 \setminus C(\epsilon,x)} d^2y \int_{T^2 \setminus \left( C(\epsilon,x)\cup C(\epsilon,y)\right)} d^2z \, w(x-y) \, w(y-z) \, w(z-x)$$ where I'm removing small disks of radius $\epsilon$ and center $x$ or $y$. Perhaps the removal of disks is not the only way of computing the integrals, just a tool.
Do you know a simple way to do it?