Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 7E (Further Complex Methods)

The poles of ${\csc }^{3}z$ inside $|z|=4$ are $z=-\pi ,0,\pi$. Near $z=k\pi$, write $z=k\pi +w$. Its Laurent series beginsso the residue is $(-1{)}^k/2$. The sum of the three residues isThe positively oriented residue theorem now gives