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Polya's shire theorem

Theorem in complex analysis From Wikipedia, the free encyclopedia

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Pólya's shire theorem, named after George Pólya, is a theorem in complex analysis that describes the asymptotic distribution of the zeros of successive derivatives of a meromorphic function on the complex plane.[1] It has applications in Nevanlinna theory.[2]:55,62

Statement

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Let be a meromorphic function on the complex plane with as its set of poles. If is the set of all zeros of all the successive derivatives , then the derived set (or the set of all limit points) is as follows:

  1. if has only one pole, then is empty.
  2. if , then coincides with the edges of the Voronoi diagram determined by the set of poles . In this case, if , the interior of each Voronoi cell consisting of the points closest to than any other point in is called the -shire.[3]

The derived set is independent of the order of each pole.[3]:32

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