The Weierstrass elliptic function is defined by

$$\mathcal{P}(z)=\frac{1}{z^{2}}+\sum_{m, n}\left[\frac{1}{\left(z-\omega_{m, n}\right)^{2}}-\frac{1}{\omega_{m, n^{2}}}\right]$$

where $\omega_{m, n}=m \omega_{1}+n \omega_{2}$, with non-zero periods $\left(\omega_{1}, \omega_{2}\right)$ such that $\omega_{1} / \omega_{2}$ is not real, and where $(m, n)$ are integers not both zero.

(i) Show that, in a neighbourhood of $z=0$,

$$\mathcal{P}(z)=\frac{1}{z^{2}}+\frac{1}{20} g_{2} z^{2}+\frac{1}{28} g_{3} z^{4}+O\left(z^{6}\right)$$

where

$$g_{2}=60 \sum_{m, n}\left(\omega_{m, n}\right)^{-4}, \quad g_{3}=140 \sum_{m, n}\left(\omega_{m, n}\right)^{-6}$$

(ii) Deduce that $\mathcal{P}$ satisfies

$$\left(\frac{d \mathcal{P}}{d z}\right)^{2}=4 \mathcal{P}^{3}-g_{2} \mathcal{P}-g_{3}$$