Let $\alpha_{1}, \alpha_{2}$ be two non-zero complex numbers with $\alpha_{1} / \alpha_{2} \notin \mathbb{R}$. Let $L$ be the lattice $\mathbb{Z} \alpha_{1} \oplus \mathbb{Z} \alpha_{2} \subset \mathbb{C}$. A meromorphic function $f$ on $\mathbb{C}$ is elliptic if $f(z+\lambda)=f(z)$, for all $z \in \mathbb{C}$ and $\lambda \in L$. The Weierstrass functions $\wp(z), \zeta(z), \sigma(z)$ are defined by the following properties:

• $\wp(z)$ is elliptic, has double poles at the points of $L$ and no other poles, and $\wp(z)=$ $1 / z^{2}+O\left(z^{2}\right)$ near 0

• $\zeta^{\prime}(z)=-\wp(z)$, and $\zeta(z)=1 / z+O\left(z^{3}\right)$ near 0 ;

• $\sigma(z)$ is odd, and $\sigma^{\prime}(z) / \sigma(z)=\zeta(z)$, and $\sigma(z) / z \rightarrow 1$ as $z \rightarrow 0$.

Prove the following

(a) $\wp$, and hence $\zeta$ and $\sigma$, are uniquely determined by these properties. You are not expected to prove the existence of $\wp, \zeta, \sigma$, and you may use Liouville's theorem without proof.

(b) $\zeta\left(z+\alpha_{i}\right)=\zeta(z)+2 \eta_{i}$, and $\sigma\left(z+\alpha_{i}\right)=k_{i} e^{2 \eta_{i} z} \sigma(z)$, for some constants $\eta_{i}, k_{i}(i=1,2)$.

(c) $\sigma$ is holomorphic, has simple zeroes at the points of $L$, and has no other zeroes.

(d) Given $a_{1}, \ldots, a_{n}$ and $b_{1}, \ldots, b_{n}$ in $\mathbb{C}$ with $a_{1}+\ldots+a_{n}=b_{1}+\ldots+b_{n}$, the function

$$\frac{\sigma\left(z-a_{1}\right) \cdots \sigma\left(z-a_{n}\right)}{\sigma\left(z-b_{1}\right) \cdots \sigma\left(z-b_{n}\right)}$$

is elliptic.

(e) $\wp(u)-\wp(v)=-\frac{\sigma(u+v) \sigma(u-v)}{\sigma^{2}(u) \sigma^{2}(v)}$.

(f) Deduce from (e), or otherwise, that $\frac{1}{2} \frac{\wp^{\prime}(u)-\wp^{\prime}(v)}{\wp(u)-\wp(v)}=\zeta(u+v)-\zeta(u)-\zeta(v)$.