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 ![P=<span style=]()
{D_1,D_2,...}" src="https://mathworld.wolfram.com/images/equations/MergelyansTheorem/Inline8.gif" style="height:15px; width:100px" /> 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.
