A function $\psi$ is defined for $z \in \mathbb{C}$ by

$$\psi(z)=\sum_{n=-\infty}^{\infty} \exp \left(\pi i\left(n+\frac{1}{2}\right)^{2} \tau+2 \pi i\left(n+\frac{1}{2}\right)\left(z+\frac{1}{2}\right)\right)$$

where $\tau$ is a complex parameter with $\operatorname{Im}(\tau)>0$. Prove that this series converges uniformly on the subsets $\{|\operatorname{Im}(z)| \leqslant R\}$ for $R>0$ and deduce that $\psi$ is holomorphic on $\mathbb{C}$.

You may assume without proof that

$$\psi(z+1)=-\psi(z) \quad \text { and } \quad \psi(z+\tau)=-\exp (-\pi i \tau-2 \pi i z) \psi(z)$$

for all $z \in \mathbb{C}$. Let $\ell(z)$ be the logarithmic derivative $\ell(z)=\frac{\psi^{\prime}(z)}{\psi(z)}$. Show that

$$\ell(z+1)=\ell(z) \quad \text { and } \quad \ell(z+\tau)=-2 \pi i+\ell(z)$$

for all $z \in \mathbb{C}$. Deduce that $\psi$ has only one zero in the parallelogram $P$ with vertices $\frac{1}{2}(\pm 1 \pm \tau)$. Find all of the zeros of $\psi .$

Let $\Lambda$ be the lattice in $\mathbb{C}$ generated by 1 and $\tau$. Show that, for $\lambda_{j}, a_{j} \in \mathbb{C}$ $(j=1, \ldots, n)$, the formula

$$f(z)=\lambda_{1} \frac{\psi^{\prime}\left(z-a_{1}\right)}{\psi\left(z-a_{1}\right)}+\ldots+\lambda_{n} \frac{\psi^{\prime}\left(z-a_{n}\right)}{\psi\left(z-a_{n}\right)}$$

gives a $\Lambda$-periodic meromorphic function $f$ if and only if $\lambda_{1}+\ldots+\lambda_{n}=0$. Deduce that $\frac{d}{d z}\left(\frac{\psi^{\prime}(z-a)}{\psi(z-a)}\right)$ is $\Lambda$-periodic.