Mergelyan,s Theorem
المؤلف:
Le Lionnais, F
المصدر:
Les nombres remarquables. Paris: Hermann
الجزء والصفحة:
...
23-7-2021
2249
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.
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