Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 3 / i / c / Solution

There are logarithmic branch points at $z=-3i$ and infinity. At zero, every local branch has $\log (3i)=\ln 3+i(\pi /2+2\pi k)\ne 0$, so division by $z^2$ gives a pole of order two at zero. Decay at infinity does not remove its branching: with $w=1/z$, the function is $w^2[\log (1+3iw)-\log w]$, which changes branch when $w$ encircles zero.