(i) Let $f(z)=\sum_{n=1}^{\infty} z^{2^{n}}$. Show that the unit circle is the natural boundary of the function element $(D(0,1), f)$.

(ii) Let $U=\{z \in \mathbf{C}: \operatorname{Re}(z)>0\} \subset \mathbf{C}$; explain carefully how a holomorphic function $f$ may be defined on $U$ satisfying the equation

$$\left(f(z)^{2}-1\right)^{2}=z$$

Let $\mathcal{F}$ denote the connected component of the space of germs $\mathcal{G}$ (of holomorphic functions on $\mathbf{C} \backslash\{0\})$ corresponding to the function element $(U, f)$, with associated holomorphic $\operatorname{map} \pi: \mathcal{F} \rightarrow \mathbf{C} \backslash\{0\}$. Determine the number of points of $\mathcal{F}$ in $\pi^{-1}(w)$ when (a) $w=\frac{1}{2}$, and (b) $w=1$.

[You may assume any standard facts about analytic continuations that you may need.]