Mergelyan's Theorem

Mergelyan's theorem can be stated as follows (Krantz 1999). Let K subset= C be compact and suppose C^*\K has only finitely many connected components. If f in C(K) is holomorphic on the interior of K and if epsilon>0, then there is a rational function r(z) with poles in C^*\K such that

 max_(z in K)|f(z)-r(z)|<epsilon.

A consequence is that if P={D_1,D_2,...} is an infinite set of disjoint open disks D_n of radius r_n such that the union is almost the unit disk. Then




Then there is a number e(P) such that M_x(P) diverges for x<e(P) and converges for x>e(P). The above theorem gives


There exists a constant which improves the inequality, and the best value known is


See also

Runge's Theorem

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Krantz, S. G. "Mergelyan's Theorem." §11.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 146-147, 1999.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 36-37, 1983.Mandelbrot, B. B. Fractals. San Francisco, CA: W. H. Freeman, p. 187, 1977.Melzack, Z. A. "On the Solid Packing Constant for Circles." Math. Comput. 23, 1969.

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Mergelyan's Theorem

Cite this as:

Weisstein, Eric W. "Mergelyan's Theorem." From MathWorld--A Wolfram Web Resource.

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