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 the year 1885. It states the following:

Denoting by C the set of complex numbers, let K be a compact subset of C 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 bounded connected component of C\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.

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