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} }} . We merely know that for all members of ( r n ) n ∈ N {\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 C ∪ { ∞ } ∖ K {\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 C {\displaystyle \mathbb {C} } and C ∖ K {\displaystyle \mathbb {C} \setminus K} is a connected set one can pick A = { ∞ } {\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 ( p n ) {\displaystyle (p_{n})} that approaches f uniformly on K.
Sketch of proof An elementary proof, inspired by Sarason (1998), proceeds as follows. There is a closed piecewise-linear contour Γ in the open set, containing K in its interior, such that all the chosen distinguished points are in its exterior. By Cauchy's integral formula
f ( w ) = 1 2 π i ∫ Γ f ( z ) d z z − w {\displaystyle f(w)={1 \over 2\pi i}\int _{\Gamma }{f(z)\,dz \over z-w}}
for w in K. Riemann approximating sums can be used to approximate the contour integral uniformly over K (there is a similar formula for the derivative). Each term in the sum is a scalar multiple of (z − w)−1 for some point z on the contour. This gives a uniform approximation by a rational function with poles on Γ. To modify this to an approximation with poles at specified points in each component of the complement of K, it is enough to check this for terms of the form (z − w)−1. If z0 is the point in the same component as z, take a path from z to z0. If two points are sufficiently close on the path, we may use the formula
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