Mergelyan's Theorem

Mergelyan's theorem can be stated as follows (Krantz 1999). Let  be compact and suppose  has only finitely many connected components. If  is holomorphic on the interior of  and if , then there is a rational function  with poles in  such that

(1)

A consequence is that if  is an infinite set of disjoint open disks  of radius  such that the union is almost the unit disk. Then

(2)

Define

(3)

Then there is a number  such that  diverges for  and converges for . The above theorem gives

(4)

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

(5)

REFERENCES:

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.