Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 3 / c / i / Solution

On the unit circle use $z=e^{i\theta }$ with $0<\theta <2\pi$. The specified branch has $\mathrm{Log}z=i\theta$ there and $dz=i{e}^{i\theta }d\theta$. ThereforeThis is the circle integral of a logarithm with a radial branch cut, using the one-sided values at the cut crossing. The logarithm is discontinuous at that crossing, so the Cauchy integral theorem for a function holomorphic around a closed contour does not apply.