# Runge theorem

Let $D$ be a simply-connected domain in the complex $z$-plane. Then any function $f$ holomorphic in $D$ can be approximated uniformly on compact sets inside $D$ by polynomials in $z$. More precisely, for any compact set $K\subset D$ and $\epsilon>0$ there is a polynomial $p(z)$ with complex coefficients such that $|f(z)-p(z)|<\epsilon$ for all $z\in K$.
In other words: Any function $f$ holomorphic in a simply-connected domain $D\subset\mathbf C$ can be represented as a series of polynomials in $z$ converging absolutely and uniformly to $f$ on compact sets inside $D$.
An equivalent statement of Runge's theorem: Let $K$ be a compact set in $\mathbf C$ with connected complement $\mathbf C\setminus K$; then any function holomorphic in a neighbourhood of $K$ can be approximated uniformly on $K$ by polynomials in $z$. In this formulation, Runge's theorem is a special case of Mergelyan's theorem (cf. Mergelyan theorem).
The following theorem on rational approximation is also called Runge's theorem: Any function $f$ holomorphic in a domain $D\subset\mathbf C$ can be uniformly approximated on compact sets inside $D$ by rational functions with poles outside $D$.