Given the Weierstrass-$\wp$ function,
$$\wp(2x+1+\tau \mid 1, \tau),$$
with half-periods $1$ and $\tau=\omega_2/ \omega_1$, I want to look at the case where $\rm{Re}(\tau) \in \mathbb{Z}$ and I want to look at the limits $\rm{Im}(\tau)>0$ both large and small. Does anyone know of any asymptotics for these two limits?
I have been looking through Abrahmowitz and Stegun but can't find anything quite useful.