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
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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
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Define
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Then there is a number such that diverges for and converges for . The above theorem gives
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There exists a constant which improves the inequality, and the best value known is
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