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

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.