Show that both the following transformations from the $z$-plane to the $\zeta$-plane are conformal, except at certain critical points which should be identified in both planes, and in each case find a domain in the $z$-plane that is mapped onto the upper half $\zeta$-plane:

$$\begin{aligned} &\text { (i) } \zeta=z+\frac{b^{2}}{z} \\ &\text { (ii) } \zeta=\cosh \frac{\pi z}{b} \end{aligned}$$

where $b$ is real and positive.