Runge's theorem

Note that not every complex number in A needs to be a pole of every rational function of the sequence

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.

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 an open set containing K, then there exists a sequence of polynomials

that approaches f uniformly on K (the assumptions can be relaxed, see Mergelyan's theorem).

Runge's theorem generalises as follows: one can take A to be a subset of the Riemann sphere C∪{∞} and require that A intersect also the unbounded connected component of K (which now contains ∞).

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.

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 two points are sufficiently close on the path, we may use the formula valid on the circle-complement

; note that the chosen path has a positive distance to K by compactness.

If z0 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.