Polynomial approximation of an exterior pole on a convex compact set ($\zeta \notin K,(z-\zeta {)}^{-1}\approx p_N(z)$)

For a compact convex set $K\subset \mathbb{C}$ and $\zeta \notin K$, strict separation gives a unit vector $u$ with $\mathrm{Re}(\bar{u}\zeta )>{\max }_K\mathrm{Re}(\bar{u}z)$. Taking $c=-Ru$ for large $R$ makes $q={\max }_K|z-c|/|\zeta -c|<1$. The polynomialhas 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.

Let $z_0$ be a boundary point of a compact convex set and choose exterior points ${\zeta }_n\to z_0$. Approximate $(z-{\zeta }_n{)}^{-1}$ uniformly on the compact set by a polynomial with error at most $1/n$. At each fixed $z\ne z_0$, the reciprocal converges and so does the approximating polynomial. On a boundary accumulating at $z_0$, uniform convergence to the singular reciprocal is impossible because each polynomial is bounded there. This distinguishes pointwise convergence from uniform convergence without confusing the two norms.