Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 4 / a / ii

For $x>0$, the upper-bank argument tends to zero, whereas the lower-bank argument tends to $2\pi$. ThusThe analytic function $f$ has identical boundary values across this part of the real axis. Hence the branch cut introduces a sign change:This compares integrand values. When two contour segments traverse opposite banks in opposite directions, their orientation supplies an additional minus sign.