A Pólya shire theorem for functions with algebraic singularities

J. K. Shaw and C. L. Prather

International Journal of Mathematics and Mathematical Sciences, 1982, vol. 5, 1-16

Abstract:

The classical shire theorem of Pólya is proved for functions with algebraic poles, in the sense of L. V. Ahlfors. A function f ( z ) is said to have an algebraic pole at z 0 provided there is a representation f ( z ) = ∑ k = − N ∞ a k ( z − z 0 ) k / p + A ( z ) , where p and N are positive integers and A ( z ) is analytic at z 0 . For p = 1 , the proof given reduces to an entirely new proof of the shire theorem. New quantitative results are given on how zeros of the successive derivatives migrate to the final set.

Date: 1982
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:861473

DOI: 10.1155/S0161171282000635

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