Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 14A / i / Solution

For $\mathrm{Re}k>0$, choose the straight path from $-i$ to $i$ and put $t=is$, $-1\le s\le 1$. Choose the power of $1-s^2$ continuously, positive for real positive $k$. The boundary factor vanishes at both endpoints. At zero,using the symmetric beta function integral. This choice also gives a nonzero solution.