Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 13B (Further Complex Methods)

A branch point of a multivalued analytic function is a point around which analytic continuation along a closed loop can return a different value. A branch cut is a curve or union of curves removed from the domain so that loops causing this monodromy are excluded and one single-valued branch can be chosen.

Write $z=r{e}^{i\theta }$ with $r>0$ and $0\le \theta <2\pi$, and write $w=u+iv$. Sincewe haveThus the inverse of the exponential is the infinitely valued complex logarithmFor the angle convention in the question, its principal value is

On taking $z$ once anticlockwise around zero, $\theta$ increases by $2\pi$ and $w$ changes by $2\pi i$; hence zero is a branch point. Under the coordinate $\zeta =1/z$, a loop around $\zeta =0$ corresponds to a loop around infinity and produces the opposite increment, so infinity is also a branch point. Removing any curve joining the two branch points prevents such loops. With $0<\arg z<2\pi$, one possible branch cut is therefore

On the cut plane choose $0<\theta <2\pi$. For $z=x+iy$,Their derivatives areThe Cauchy-Riemann equations hold, so this branch is holomorphic. Its derivative is