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 C ∪ { ∞ } {\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 C ∪ { ∞ } ∖ K {\displaystyle \mathbb {C} \cup \{\infty \}\setminus K} , then there exists a sequence ( r n ) n ∈ N {\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 ( r n ) n ∈ N {\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 ( r n ) n ∈ N {\displaystyle (r_{n})_{n\in \mathbb {N} }} .