Let $\Lambda$ be a lattice in $\mathbb{C}$ generated by 1 and $\tau$, where $\operatorname{Im} \tau>0$. The Weierstrass function $\wp$ is the unique meromorphic $\Lambda$-periodic function on $\mathbb{C}$, such that the only poles of $\wp$ are at points of $\Lambda$ and $\wp(z)-1 / z^{2} \rightarrow 0$ as $z \rightarrow 0$.

Show that $\wp$ is an even function. Find all the zeroes of $\wp^{\prime}$.

Suppose that $a$ is a complex number such that $2 a \notin \Lambda$. Show that the function

$$h(z)=(\wp(z-a)-\wp(z+a))(\wp(z)-\wp(a))^{2}-\wp^{\prime}(z) \wp^{\prime}(a)$$

has no poles in $\mathbb{C} \backslash \Lambda$. By considering the Laurent expansion of $h$ at $z=0$, or otherwise, deduce that $h$ is constant.

[General properties of meromorphic doubly-periodic functions may be used without proof if accurately stated.]