Let $\ell(z)$ be an analytic branch of $\log z$ on a domain $D \subset \mathbb{C} \backslash\{0\}$. Write down an analytic branch of $z^{1 / 2}$ on $D$. Show that if $\psi_{1}(z)$ and $\psi_{2}(z)$ are two analytic branches of $z^{1 / 2}$ on $D$, then either $\psi_{1}(z)=\psi_{2}(z)$ for all $z \in D$ or $\psi_{1}(z)=-\psi_{2}(z)$ for all $z \in D$.

Describe the principal value or branch $\sigma_{1}(z)$ of $z^{1 / 2}$ on $D_{1}=\mathbb{C} \backslash\{x \in \mathbb{R}: x \leqslant 0\}$. Describe a branch $\sigma_{2}(z)$ of $z^{1 / 2}$ on $D_{2}=\mathbb{C} \backslash\{x \in \mathbb{R}: x \geqslant 0\}$.

Construct an analytic branch $\varphi(z)$ of $\sqrt{1-z^{2}}$ on $\mathbb{C} \backslash\{x \in \mathbb{R}:-1 \leqslant x \leqslant 1\}$ with $\varphi(2 i)=\sqrt{5}$. [If you choose to use $\sigma_{1}$ and $\sigma_{2}$ in your construction, then you may assume without proof that they are analytic.]

Show that for $0<|z|<1$ we have $\varphi(1 / z)=-i \sigma_{1}\left(1-z^{2}\right) / z$. Hence find the first three terms of the Laurent series of $\varphi(1 / z)$ about 0 .

Set $f(z)=\varphi(z) /\left(1+z^{2}\right)$ for $|z|>1$ and $g(z)=f(1 / z) / z^{2}$ for $0<|z|<1$. Compute the residue of $g$ at 0 and use it to compute the integral

$$\int_{|z|=2} f(z) d z$$