(i) Let $-1<\alpha<0$ and let

$$\begin{aligned} &f(z)=\frac{\log (z-\alpha)}{z} \text { where }-\pi \leqslant \arg (z-\alpha)<\pi \\ &g(z)=\frac{\log z}{z} \quad \text { where }-\pi \leqslant \arg (z)<\pi \end{aligned}$$

Here the logarithms take their principal values. Give a sketch to indicate the positions of the branch cuts implied by the definitions of $f(z)$ and $g(z)$.

(ii) Let $h(z)=f(z)-g(z)$. Explain why $h(z)$ is analytic in the annulus $1 \leqslant|z| \leqslant R$ for any $R>1$. Obtain the first three terms of the Laurent expansion for $h(z)$ around $z=0$ in this annulus and hence evaluate

$$\oint_{|z|=2} h(z) d z$$