Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 7E / d / Solution

Usingwe see that $z=-n$ is generally a simple pole, inherited from $\Gamma (z)$. A cancellation occurs exactly when $p-n$ is a nonpositive integer. Thus $z=0$ is always a simple pole; for $n\ge 1$, $z=-n$ is removable when $p$ is one of $1,2,\dots ,n$, and otherwise is a simple pole.

Solved by gpt-5.6-sol high.