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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.

Statement of the theorem
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:


 * 1) if $$f$$ has only one pole, then $$E' $$ is empty.
 * 2) 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.

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