= Solution
For an <isolated singularity> at $a$, write the <Laurent expansion> in a punctured disk as $f(z)=\sum_{k=-\infty}^{\infty}c_k(z-a)^k$. Its <residue> is
$$
\boxed{\operatorname{res}(f;a)=c_{-1}.}
$$
The <sine> in the denominator has simple zeros precisely at the integers, and the numerator is nonzero there. Consequently $g(z)=\pi\cot(\pi z)$ has <simple poles> exactly at $n\in\mathbb Z$, with
$$
\boxed{\operatorname{res}(g;n)=\frac{\pi\cos(\pi n)}{\pi\cos(\pi n)}=1.}
$$
There are no other finite <singularities>.
Back to article page
