(i) Explain why the formula

$$f(z)=\sum_{n=-\infty}^{\infty} \frac{1}{(z-n)^{2}}$$

defines a function that is analytic on the domain $\mathbb{C} \backslash \mathbb{Z}$. [You need not give full details, but should indicate what results are used.]

Show also that $f(z+1)=f(z)$ for every $z$ such that $f(z)$ is defined.

(ii) Write $\log z$ for $\log r+i \theta$ whenever $z=r e^{i \theta}$ with $r>0$ and $-\pi<\theta \leqslant \pi$. Let $g$ be defined by the formula

$$g(z)=f\left(\frac{1}{2 \pi i} \log z\right)$$

Prove that $g$ is analytic on $\mathbb{C} \backslash\{0,1\}$.

[Hint: What would be the effect of redefining $\log z$ to be $\log r+i \theta$ when $z=r e^{i \theta}$, $r>0$ and $0 \leqslant \theta<2 \pi$ ?]

(iii) Determine the nature of the singularity of $g$ at $z=1$.