 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

A <pole> of order $m\geq1$ at $a$ means
$$
\boxed{f(z)=\frac{g(z)}{(z-a)^m},\qquad g\text{ holomorphic near }a,\quad g(a)\ne0.}
$$
Equivalently the lowest nonzero power in its <Laurent series> is $(z-a)^{-m}$. In particular, $(z-a)^mf(z)$ tends to a finite nonzero limit.

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