Runge's theorem

In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in 1885. It states the following:

Denoting by C the set of complex numbers, let K be a closed subset of ${\displaystyle \mathbb {C} \cup \{\infty \}}$ and let f be a function which is holomorphic on an open set containing K. If A is a set containing at least one complex number from every connected component of ${\displaystyle \mathbb {C} \cup \{\infty \}\setminus K}$, then there exists a sequence ${\displaystyle (r_{n})_{n\in \mathbb {N} }}$ of rational functions which converges uniformly to f on K and such that all the poles of the functions ${\displaystyle (r_{n})_{n\in \mathbb {N} }}$ are in A.

Note that not every complex number in A needs to be a pole of every rational function of the sequence ${\displaystyle (r_{n})_{n\in \mathbb {N} }}$. We merely know that for all members of ${\displaystyle (r_{n})_{n\in \mathbb {N} }}$ that do have poles, those poles lie in A.

One aspect that makes this theorem so powerful is that one can choose the set A arbitrarily. In other words, one can choose any complex numbers from the bounded connected components of ${\displaystyle \mathbb {C} \cup \{\infty \}\setminus K}$ and the theorem guarantees the existence of a sequence of rational functions with poles only amongst those chosen numbers.

For the special case in which K is a compact subset of ${\displaystyle \mathbb {C} }$ and ${\displaystyle \mathbb {C} \setminus K}$ is a connected set one can pick ${\displaystyle A=\{\infty \}}$. Since rational functions with no poles except at infinity are simply polynomials, we get the following corollary: If K is a compact subset of C such that C\K is a connected set, and f is a holomorphic function on an open set containing K, then there exists a sequence of polynomials ${\displaystyle (p_{n})}$ that approaches f uniformly on K.