Runge's theorem
= Runge's theorem
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= Runge theorem
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= Runge approximation theorem
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Let $K\subset\mathbb C$ be <compact>, and let $f$ be <holomorphic> on a neighborhood of $K$. Choose a set of allowed poles in the <Riemann sphere> outside $K$, meeting every connected component of its complement. Then rational functions with poles only in that allowed set approximate $f$ uniformly on $K$. The point at infinity may serve as the pole in the unbounded component.
If $\mathbb C\setminus K$ is connected, one can allow only infinity, giving the <polynomial Runge theorem>. If $K$ is the <unit circle>, its complement has an inside and an outside component. The function $1/z$ is holomorphic near $K$ and is already a rational approximant with its pole in the hole; it cannot be uniformly approximated there by polynomials, since their <contour integrals> vanish while $\oint dz/z=2\pi i$.
The proof first approximates a <Cauchy integral> by a finite rational sum with nearby poles. The <pole-moving polynomial approximation> argument transports poles through complementary components. This explains why complement geometry, rather than just smoothness of $f$, governs the allowed approximation class.