International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 4, Pages 691-706

A Pólya “shire” theorem for functions with algebraic singularities

J. K. Shaw and C. L. Prather

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg 24061-4097, Virginia, USA

Received 5 February 1982

Copyright © 1982 J. K. Shaw and C. L. Prather. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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=Nak(zz0)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.