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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.
ISSN 0305-0041

**Full text not available from this repository.**

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 |
---|---|

Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Science > Mathematics |

Journal or Publication Title: | Mathematical Proceedings of the Cambridge Philosophical Society |

Publisher: | Cambridge University Press |

ISSN: | 0305-0041 |

Official Date: | September 2008 |

Volume: | Vol.145 |

Number: | No.2 |

Number of Pages: | 16 |

Page Range: | pp. 257-272 |

Identification Number: | 10.1017/S0305004108001643 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

Funder: | EPSRC |

Grant number: | RA22AP |

URI: | http://wrap.warwick.ac.uk/id/eprint/48353 |

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