# Runge theorem

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A theorem on the possibility of polynomially approximating holomorphic functions, first proved by C. Runge (1885) (cf. also Approximation of functions of a complex variable).

Let be a simply-connected domain in the complex -plane. Then any function holomorphic in can be approximated uniformly on compact sets inside by polynomials in . More precisely, for any compact set and there is a polynomial with complex coefficients such that for all .

In other words: Any function holomorphic in a simply-connected domain can be represented as a series of polynomials in converging absolutely and uniformly to on compact sets inside .

An equivalent statement of Runge's theorem: Let be a compact set in with connected complement ; then any function holomorphic in a neighbourhood of can be approximated uniformly on by polynomials in . In this formulation, Runge's theorem is a special case of Mergelyan's theorem (cf. Mergelyan theorem).

The following theorem on rational approximation is also called Runge's theorem: Any function holomorphic in a domain can be uniformly approximated on compact sets inside by rational functions with poles outside .

Runge's theorem has many applications in the theory of functions of a complex variable and in functional analysis. A theorem analogous to Runge's theorem is valid for non-compact Riemann surfaces. An extension of Runge's theorem to functions of several complex variables is the Oka–Weil theorem (see Oka theorems).

#### References

 [1] A.I. Markushevich, "A short course on the theory of analytic functions" , Moscow (1978) (In Russian) [2] B.V. Shabat, "Introduction of complex analysis" , 1 , Moscow (1976) (In Russian)