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# Runge's Theorem

Let be compact, let be analytic on a neighborhood of , and let contain at least one point from each connected component of . Then for any , there is a rational function with poles in such that

(Krantz 1999, p. 143).

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

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

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## References

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.

Runge's Theorem

## Cite this as:

Weisstein, Eric W. "Runge's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RungesTheorem.html