Consider a multi-valued function $w(z)$.

(a) Explain what is meant by a branch point and a branch cut.

(b) Consider $z=e^{w}$.

(i) By writing $z=r e^{i \theta}$, where $0 \leqslant \theta<2 \pi$, and $w=u+i v$, deduce the expression for $w(z)$ in terms of $r$ and $\theta$. Hence, show that $w$ is infinitely valued and state its principal value.

(ii) Show that $z=0$ and $z=\infty$ are the branch points of $w$. Deduce that the line $\operatorname{Im} z=0, \operatorname{Re} z>0$ is a possible choice of branch cut.

(iii) Use the Cauchy-Riemann conditions to show that $w$ is analytic in the cut plane. Show that $\frac{d w}{d z}=\frac{1}{z}$.