Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 3 / 5D / i

Take the circle with the usual positive, counterclockwise orientation. The only singularity is at zero, and the Laurent series givesThe coefficient of $z^{-1}$ is $1/2$, so the residue theorem givesThe essential singularity at zero still has a well-defined residue; the theorem does not require it to be a pole.