Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 12B (Complex Analysis or Complex Methods)

The partial fraction decomposition isFor $0<|z|<2$, expand the second term as a geometric series:Hence

For $|z|>2$, expand directly in inverse powers:Therefore the outer Laurent series is

Put $w=z-1$. ThenThusThis is actually a Taylor series, because $z=1$ is a regular point. The point $z=0$ is a simple pole. At infinity,extends analytically to $w=0$, so infinity is a removable singularity of $f$, with extended value zero.

For the real integral, first writeLetWith $z=e^{i\theta }$,The denominator has roots $1/2$ and $2$, so only $1/2$ lies inside the contour. Its residue is $1/(3i)$, and the residue theorem givesConsequently