Draft:Polya's shire theorem

Submission declined on 5 April 2024 by Pbritti (talk).

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Declined by Pbritti 2 months ago. Last edited by Citation bot 60 days ago. Reviewer: Inform author.

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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 $f$ be a meromorphic function on the complex plane with $P\neq \emptyset$ as its set of poles. If $E$ is the set of all zeros of all the successive derivatives $f',f'',f^{(3)},\ldots$ , then the derived set $E'$ (or the set of all limit points) is as follows:

if $f$ has only one pole, then $E'$ is empty.
if $|P|\geq 2$ , then $E'$ coincides with the edges of the Voronoi diagram determined by the set of poles $P$ . In this case, if $a\in P$ , the interior of each Voronoi cell consisting of the points closest to $a$ than any other point in $P$ is called the $a$ -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.{{cite book}}: CS1 maint: date and year (link)

[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)

[1]

[2]