Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 4 / 13E / c / Solution

The image contour has polar representation $2\cos 2\theta e^{3i\theta }$. It is invariant under complex conjugation and under reflection through the origin. As $\theta$ runs from zero to $2\pi$, the successive real-axis visits areat angles $0,\pi /4,\pi /3,2\pi /3,3\pi /4,\pi ,5\pi /4,4\pi /3,5\pi /3,7\pi /4,2\pi$. To specify the crossing direction, differentiate:Its imaginary component is $6$ at the point $2$, $-6$ at $-2$, $3$ at either visit to $1$, and $-3$ at either visit to $-1$. At a visit to zero it is $-4\cos \theta$, nonzero for each of the four angles. Therefore all these visits are transverse crossings of the real axis, even though different arcs share the same image point.