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).