= Polynomial approximation of an exterior pole on a convex compact set
{title2=$\zeta\notin K,\quad (z-\zeta)^{-1}\approx p_N(z)$}
For a compact <convex set> $K\subset\mathbb C$ and $\zeta\notin K$, strict separation gives a unit vector $u$ with $\operatorname{Re}(\bar u\zeta)>\max_K\operatorname{Re}(\bar uz)$. Taking $c=-Ru$ for large $R$ makes $q=\max_K|z-c|/|\zeta-c|<1$. The polynomial
$$
p_N(z)=-\frac1{\zeta-c}\sum_{j=0}^N\left(\frac{z-c}{\zeta-c}\right)^j
$$
has uniform error at most $q^{N+1}/(|\zeta-c|(1-q))$. This elementary <geometric series> argument proves exterior-pole approximation without a general complex approximation theorem.
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