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In complex analysis, Runge's theorem, also known as Runge's approximation theorem, named after the German Mathematician Carl Runge, and put forward by him in the year 1885, states the following: If K is a compact subset of C (the set of complex numbers), A is a set containing at least one complex number from every bounded connected component of CK, and f is a holomorphic function on K, then there exists a sequence $(r\_n)$ of rational functions with poles in A such that the sequence $(r\_n)$ approaches the function f uniformly on K.

Notice that not every complex number in A need be a pole of every rational function of the sequence $(r\_n)$. We merely know that if some $r\_n$ of the sequence has poles, those poles are in A.

One of the things that makes this theorem so powerful is that one can choose the set A at will. In other words, one can pick any complex numbers as one wishes from the bounded connected components of CK. Then the theorem guarantees the existence of a sequence of rational functions with poles only in those chosen numbers.

In the special case that CK is a connected set, the set A in the theorem will clearly be empty. And since rational functions with no poles are indeed nothing but polynomials, we get the following corollary: If K is a compact subset of C such that CK is a connected set, and f is a holomorphic function on K, then there exists a sequence of polynomials $(p\_n)$ that approaches f uniformly on K.

A slightly more general version of this theorem is obtained if one takes A to be a subset of the Riemann sphere C∪{∞} and then requires A to intersect also the unbounded connected component of K (which now contains ∞). That is, in the formulation given above, the rational functions may turn out to have a pole at infinity, while in the more general formulation the pole can be chosen instead anywhere in the unbounded connected component of K.

- John B. Conway, A Course in Functional Analysis, Springer; 2 edition (1997), ISBN 0-387-97245-5.
- Robert E. Greene and Steven G. Krantz, Function Theory of One Complex Variable, American Mathematical Society; Second Edition (2002), ISBN 0-8218-2905-X.

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Last updated on Saturday July 12, 2008 at 17:39:48 PDT (GMT -0700)

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Last updated on Saturday July 12, 2008 at 17:39:48 PDT (GMT -0700)

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