Runge's Theorem

Let K subset= C be compact, let f be analytic on a neighborhood of K, and let P subset= C^*\K contain at least one point from each connected component of C^*\K. Then for any epsilon>0, there is a rational function r(z) with poles in P such that

 max_(z in K)|f(z)-r(z)|<epsilon

(Krantz 1999, p. 143).

A polynomial version can be obtained by taking P={infty}. Let f(x) be an analytic function which is regular in the interior of a Jordan curve C and continuous in the closed domain bounded by C. Then f(x) can be approximated with arbitrary accuracy by polynomials (Szegö 1975, p. 5; Krantz 1999, p. 144).

See also

Analytic Function, Jordan Curve, Mergelyan's Theorem, Regular Function

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Krantz, S. G. "Runge's Theorem." §11.1.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 143-144, 1999.Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 7, 1975.

Referenced on Wolfram|Alpha

Runge's Theorem

Cite this as:

Weisstein, Eric W. "Runge's Theorem." From MathWorld--A Wolfram Web Resource.

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