Draft:Polya's shire theorem
Submission declined on 5 April 2024 by Pbritti (talk). Neologisms are not considered suitable for Wikipedia unless they receive substantial use and press coverage; this requires strong evidence in independent, reliable, published sources. Links to sites specifically intended to promote the neologism itself do not establish its notability.
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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
[edit]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:
- if has only one pole, then is empty.
- 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
[edit]- ^ Pólya, George (1922). "Über die Nullstellen sukzessiver Derivierten". Math. Zeit. 12: 36–60. doi:10.1007/BF01482068.
- ^ Whittaker, J.M. (1935). Interpolatory Function Theory. Cambridge University Press. pp. 32–38.
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