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Cercles de Remplissage and Asymptotic Behaviour

Published online by Cambridge University Press:  20 November 2018

Paul Gauthier
Paul Gauthier*
Affiliation:
Université de Montréal, Montréal, Québec
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In 1908, Lindelöf showed that if w = f(x) is a bounded holomorphic function in a sector S: |arg z| < θ1, and if f(z) has an asymptotic value w0 as z tends to ∞ along a half-ray in S; then f(z) tends uniformly to w0 as z tends to ∞o within any sector |arg z| ≦ θ, 0 ≦ θ < θ1. Montel (8) later replaced the condition that f(z) be bounded by the condition that f(z) be meromorphic and omitted three values. The following is an immediate consequence of the Lindelöf-Montel theorem.

THEOREM 1. Let w = f(x) be a function meromorphic in the sector | arg z| < θ1, and let f(z) tend to a value w0 as z → ∞ along the positive real axis.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 447 - 455
Copyright
Copyright © Canadian Mathematical Society 1969

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