Draft:Polya's shire theorem

  • Comment: While I believe the references present accurately source this information, I do not see indication that the theorem is notable. Consider merging to the article on George Pólya. Pbritti (talk) 20:23, 5 April 2024 (UTC)

In complex analysis, Pólya's shire theorem, due to the mathematician George Pólya, describes the asymptotic distribution of the zeros of successive derivatives of a meromorphic function on the complex plane.[1]

Statement of the theorem

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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.[2]

The derived set is independent of the location or order of each pole.

References

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  1. ^ Pólya, George (1922). "Über die Nullstellen sukzessiver Derivierten". Math. Zeit. 12: 36–60. doi:10.1007/BF01482068.
  2. ^ Whittaker, J.M. (1935). Interpolatory Function Theory. Cambridge University Press. pp. 32–38.{{cite book}}: CS1 maint: date and year (link)