Runge Theory for Compact Sets

Abstract

In discs B, all holomorphic functions are approximated compactly by their Taylor polynomials. In particular, for every f ∈ O(B) and every compact set K in B, there exists a sequence of polynomials p_nsuch that lim |f - P_n|K = 0. In arbitrary domains, polynomial approximation is not always possible; in C^×, for example, there is no sequence of polynomials p_n that approximates the holomorphic function l/z uniformly on a circle γ, for it would then follow that

$$$$2\pi i = \int_\gamma{\frac{{d\varsigma }}{\varsigma }}= \lim \int_\gamma{{p_n}\left( \varsigma\right)d\varsigma }= 0$$

Runge approximation theory charms by its wonderful balance between freedom and necessity.