Runge’s theorem

Let $K$ be a compact subset of $\mathbb{C}$ , and let $E$ be a subset of $\mathbb{C}_{\infty}=\mathbb{C}\cup\{\infty\}$ (the extended complex plane) which intersects every connected component of $\mathbb{C}_{\infty}-K$ . If $f$ is an analytic function in an open set containing $K$ , given $\varepsilon>0$ , there is a rational function $R(z)$ whose only poles are in $E$ , such that $|f(z)-R(z)|<\varepsilon$ for all $z\in K$ .

Title	Runge’s theorem
Canonical name	RungesTheorem
Date of creation	2013-03-22 13:15:12
Last modified on	2013-03-22 13:15:12
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	6
Author	Koro (127)
Entry type	Theorem
Classification	msc 30E10
Related topic	MergelyansTheorem