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).

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.