Mergelyan’s theorem

Theorem (Mergelyan).

Let KC be a compact subset of the complex plane such that C\K (the complement of K) is connectedPlanetmathPlanetmath, and let f:KC be a continuous functionMathworldPlanetmath which is also holomorphic on the interior of K. Then f is the uniform limit on K of holomorphic polynomials (polynomials in one complex variable).

So for any ϵ>0 one can find a polynomial p(z)=j=1najzj such that |f(z)-p(z)|<ϵ for all zK.

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


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
  • 2 Walter Rudin. . McGraw-Hill, Boston, Massachusetts, 1987.
Title Mergelyan’s theorem
Canonical name MergelyansTheorem
Date of creation 2013-03-22 14:23:59
Last modified on 2013-03-22 14:23:59
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Theorem
Classification msc 30E10
Related topic RungesTheorem