Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 1 / 3 (Complex Analysis OR Complex Methods)

The residue is the coefficient $a_{-1}$ in the Laurent expansion. At a double pole, $h(z)=(z-z_0{)}^{2}f(z)$ is holomorphic and $h(z)=h(z_0)+h\prime (z_0)(z-z_0)+\cdots$, so $\mathrm{Res}(f,z_0)=h\prime (z_0)$.

Partial fractions giveFor $1<|z|<2$ this becomes

At zero: $\sin z$ is holomorphic with a zero; $1/\sin z$ has a simple pole of residue $1$; $z/\sin z$ has a removable singularity with value $1$; $\sin (1/z)$ is essential with residue $1$; $z\sin (1/z)$ is essential with residue $0$. Finally $1/\sin (1/z)$ has poles accumulating at zero, so zero is a non-isolated singularity and no residue there is defined.