Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 12 / 12.1G

For any circle $|\zeta |=\rho$ with $r<\rho <R$, Cauchy's formula applied inside and outside the circle gives the locally uniformly convergent Laurent expansionAt zero, the singularity is removable exactly when all $a_n$ with $n<0$ vanish; it is a pole of order $k$ when $a_{-k}\ne 0$ and $a_n=0$ for $n<-k$; it is essential when infinitely many negative coefficients are nonzero.

Since ${\csc }^{2}z=z^{-2}+1/3+O(z^2)$, the principal parts cancel and $f_1=-1/3+O(z^2)$. The singularity is removable.

Termwise integration of the exponential series givesThere are infinitely many negative Laurent coefficients, so zero is an essential singularity.