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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Complex iterated radicals


Author: Leon Gerber
Journal: Proc. Amer. Math. Soc. 41 (1973), 205-210
MSC: Primary 40A05
MathSciNet review: 0318721
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Abstract: We prove the convergence of the sequence $ S$ defined by $ {z_{n + 1}} = {({z_n} - c)^{1/2}},c$ real, for any choice of $ {z_0}$. Let $ k = \vert\tfrac{1}{4} - c{\vert^{1/2}}$. If $ c < 0$ or $ c = \tfrac{1}{4},S$ has only one fixed point $ w = \tfrac{1}{2} + k$ and converges to $ w$ for any $ {z_0}$. If $ 0 \leqq c < \tfrac{1}{4},S$ has the fixed points $ {w_1} = \tfrac{1}{2} + k$ and $ {w_2} = \tfrac{1}{2} - k$, and for any $ {z_0} \ne {w_2},S$ converges to $ {w_1}$. If $ c > \tfrac{1}{4},S$ has the fixed points $ {w_1} = \tfrac{1}{2} + ik$ and $ {w_2} = \tfrac{1}{2} - ik$ and converges to $ {w_1}$ if $ \operatorname{Re} ({z_0}) \geqq 0$ and to $ {w_2}$ otherwise. We show that convergence is strictly monotone when the neighborhood system is the pencil of coaxial circles with $ {w_1}$ and $ {w_2}$ as limiting points, and give rates of convergence.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0318721-1
PII: S 0002-9939(1973)0318721-1
Keywords: Iterated radicals, convergence, rate of convergence, coaxial circles
Article copyright: © Copyright 1973 American Mathematical Society