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Date: 20-5-2021
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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.
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