# Runge’s theorem

Let $K$ be a compact subset of $\mathbb{C}$, and let $E$ be a subset of $\mathbb{C}_{\infty}=\mathbb{C}\cup\{\infty\}$ (the extended complex plane) which intersects every connected component of $\mathbb{C}_{\infty}-K$. If $f$ is an analytic function in an open set containing $K$, given $\varepsilon>0$, there is a rational function $R(z)$ whose only poles are in $E$, such that $|f(z)-R(z)|<\varepsilon$ for all $z\in K$.

Title Runge’s theorem RungesTheorem 2013-03-22 13:15:12 2013-03-22 13:15:12 Koro (127) Koro (127) 6 Koro (127) Theorem msc 30E10 MergelyansTheorem