PlanetMath: Mergelyan's theorem

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Mergelyan's theorem

(Theorem)

Theorem 1 (Mergelyan) Let $K \subset {\mathbb{C}}$ be a compact subset of the complex plane such that ${\mathbb{C}} \backslash K$ (the complement of ) is connected, and let $f\colon K \to {\mathbb{C}}$ be a continuous function which is also holomorphic on the interior of Then is the uniform limit on of holomorphic polynomials (polynomials in one complex variable).

So for any $\epsilon > 0$ one can find a polynomial $p(z) = \sum_{j=1}^n a_j z^j$ such that $\lvert f(z) - p(z) \rvert < \epsilon$ for all $z \in K.$

Do note that this theorem is not a weaker version of Runge's theorem. Here, we do not need to be holomorphic on a neighbourhood of but just on the interior of For example, if the interior of is empty, then just needs to be continuous on Further, it could be that the closure of the interior of might not be all of Consider $K = D \cup [-10,10],$ where is the closed unit disc. Then has two lines coming out of either end of the disc and needs to only be continuous there.

Also note that this theorem is distinct from the Stone-Weierstrass theorem. The point here is that the polynomials are holomorphic in Mergelyan's theorem.

Bibliography

1: John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
2: Walter Rudin. Real and Complex Analysis. McGraw-Hill, Boston, Massachusetts, 1987.

"Mergelyan's theorem" is owned by jirka.

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