Past exam of the metric spaces and complex analysis course of the University of Oxford / 2021 / 5 / a / i

Start with the partial fraction decompositionFor $1<|z|<2$, expand the first fraction in negative powers and the second in nonnegative powers using the geometric series:Thus the Laurent expansion on this annulus isFor $|z|>2$, both fractions expand in negative powers:Combining the two geometric series givesThe $z^{-1}$ coefficient is zero, consistently with the original rational function being $O(z^{-2})$ at infinity. Each displayed series converges absolutely on its stated annulus and uniformly on every compact subset of that annulus.