Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 14B / ii / Solution

Contract the loops in the Pochhammer contour onto the interval $(0,1)$. When $\mathrm{Re}z,\mathrm{Re}b>0$, the small-circle contributions tend to zero. The two clockwise circuits change the branches by $e^{-2\pi ib}$ and $e^{-2\pi iz}$, and the return circuits undo them. The four interval contributions sum toUsing the Beta function and $1-e^{-2\pi iu}=2i{e}^{-\pi iu}\sin (\pi u)$ gives