A Pólya “shire” theorem for functions with algebraic singularities

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 z0 provided there is a representation f(z)=∑k=−N∞ak(z−z0)k/p+A(z), where p and N are positive integers and A(z) is analytic at z0. 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.