Arakelyan's theorem

In mathematics, Arakelyan's theorem is a generalization of Mergelyan's theorem from compact subsets of an open subset of the complex plane to relatively closed subsets of an open subset.

Theorem

Let Ω be an open subset of and E a relatively closed subset of Ω. By Ω^* is denoted the Alexandroff compactification of Ω.

Arakelyan's theorem states that for every f continuous in E and holomorphic in the interior of E and for every ε > 0 there exists g holomorphic in Ω such that |g&nbsp;−&nbsp;f|&nbsp;<&nbsp;ε on E if and only if Ω^*&nbsp;\&nbsp;E is connected and locally connected.

See also

• Runge's theorem
• Mergelyan's theorem

References