# 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 of rational functions which converges uniformly to *f* on *K* and such that all the poles of the functions are in *A.*

Note that not every complex number in *A* needs to be a pole of every rational function of the sequence . We merely know that for all members of 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* 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 **C**\*K* is a connected set (or equivalently that *K* is simply-connected), the set *A* in the theorem will clearly be empty. Since rational functions with no poles 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 *K*, then there exists a sequence of polynomials that approaches *f* uniformly on *K*.

Runge's theorem generalises as follows: if one takes *A* to be a subset of the Riemann sphere **C**∪{∞} and requires that *A* 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*.

## Proof

An elementary proof, given in Template:Harvtxt, proceeds as follows. There is a closed piecewise-linear contour Γ in the open set, containing *K* in its interior. By Cauchy's integral theorem

for *w* in *K*. Riemann approximating sums can be used to approximate the contour integral uniformly over *K*. 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 *z*_{0} is the point in the same component as *z*, take a piecewise-linear path from *z* to *z*_{0}. If two points are sufficiently close on the path, any rational function with poles only at the first point can be expanded as a Laurent series about the second point. That Laurent series can be truncated to give a rational function with poles only at the second point uniformly close to the original function on *K*. Proceeding by steps along the path from *z* to *z*_{0} the original function (*z* − *w*)^{−1} can be successively modified to give a rational function with poles only at *z*_{0}.

If *z*_{0} is the point at infinity, then by the above procedure the rational function (*z* − *w*)^{−1} can first be approximated by a rational function *g* with poles at *R* > 0 where *R* is so large that *K* lies in *w* < *R*. The Taylor series expansion of *g* about 0 can then be truncated to give a polynomial approximation on *K*.

## See also

## References

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