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Gaussian integer points of analytic functions in a half-plane
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Fletcher, Alastair (2008) Gaussian integer points of analytic functions in a half-plane. Mathematical Proceedings of the Cambridge Philosophical Society, Vol.145 (No.2). pp. 257-272. doi:10.1017/S0305004108001643 ISSN 0305-0041.
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Official URL: http://dx.doi.org/10.1017/S0305004108001643
Abstract
A classical result of Pólya states that 2z is the slowest growing transcendental entire function taking integer values on the non-negative integers. Langley generalised this result to show that 2z is the slowest growing transcendental function in the closed right half-plane Ω = {z xs2208 : Re(z) ≥ 0} taking integer values on the non-negative integers. Let E be a subset of the Gaussian integers in the open right half-plane with positive lower density and let f be an analytic function in Ω taking values in the Gaussian integers on E. Then in this paper we prove that if f does not grow too rapidly, then f must be a polynomial. More precisely, there exists L > 0 such that if either the order of growth of f is less than 2 or the order of growth is 2 and the type is less than L, then f is a polynomial.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Mathematical Proceedings of the Cambridge Philosophical Society | ||||
Publisher: | Cambridge University Press | ||||
ISSN: | 0305-0041 | ||||
Official Date: | September 2008 | ||||
Dates: |
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Volume: | Vol.145 | ||||
Number: | No.2 | ||||
Number of Pages: | 16 | ||||
Page Range: | pp. 257-272 | ||||
DOI: | 10.1017/S0305004108001643 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Funder: | EPSRC | ||||
Grant number: | RA22AP |
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