Newton's method on the complex exponential function
Author:
Mako E. Haruta
Journal:
Trans. Amer. Math. Soc. 351 (1999), 24992513
MSC (1991):
Primary 58F23
Published electronically:
February 15, 1999
MathSciNet review:
1422898
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We show that when Newton's method is applied to the product of a polynomial and the exponential function in the complex plane, the basins of attraction of roots have finite area.
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F. Przytycki, Remarks on the Simple Connectedness of Basins of Sinks for Iterations of Rational Maps, Preprint, Polish Academy of Sciences, Warsaw, 1987.
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S. Sutherland, Finding Roots of Complex Polynomials with Newton's Method, (preprint) Institute for Mathematical Sciences, SUNY Stony Brook, 1990.
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 D. Alexander, The Historical Background to the Works of Pierre Fatou and Gaston Julia in Complex Dynamics, Thesis, Boston University, 1992.
 [Ber]
 W. Bergweiler, et al., Newton's Method for Meromorphic Functions, preprint, Lehrstuhl C für Mathematik, RheinischWestfälische Technische Hochschule, Aachen, 1991.
 [BSV]
 P. Blanchard, S. Sutherland, G. Vegter, Citool, computer software, Boston University Mathematics Department, 1986.
 [C3]
 A. Cayley, Application of the NewtonFourier Method to an Imaginary Root of an Equation, Quart. J. of Pure and Applied Math. XVI (1879), 179185.
 [CGS]
 J. Curry, L. Garnett and D. Sullivan, On the iteration of a rational function: Computer experiments with Newton's method, Comm. Math. Phys. 91(1983), 267277. MR 85e:30040
 [H]
 M. Hurley, Multiple Attractors in Newton's Method, Ergodic Theory Dynamical Systems 6 (1986), 561569. MR 88a:58123
 [HK1]
 F. von Haeseler and H. Kriete, The Relaxed Newton's Method for Polynomials, preprint, Institut für Dynamische Systeme, Universität Bremen, 1990.
 [HM]
 M. Hurley and C. Martin, Newton's Algorithm and Chaotic Dynamical Systems, SIAM J. Math. Anal. 15 (1984), 238. MR 85j:58101
 [HP]
 F. von Haeseler and H.O. Peitgen, Newton's method and complex dynamical systems, Acta Appl. Math. 13 (1988), 358. MR 96a:58102
 [Kr]
 H. Kreite, A Newton's Method Case with a Basin of Infinite Area, Preprint, Fakultät und Institut für Mathematik, RuhrUniversität Bochum, 1992.
 [M]
 J. Milnor, Dynamics in One Complex Dimension: Introductory Lectures, preprint #1990/5, SUNY StonyBrook, Institute for Mathematical Sciences.
 [Ma]
 A. Manning, How to be Sure of Solving a Complex Polynomial using Newton's Method, preprint, Mathematics Institute, University of Warwick, 1986.
 [Mc]
 C. McMullen, Families of rational maps and iterative rootfinding algorithms, Ann. of Math. 125 (1987), 467493. MR 88i:58082
 [MSS]
 R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. sci. Éc. Norm. Sup., 4e série, 16 (1983), 193217. MR 85j:58089
 [P]
 F. Przytycki, Remarks on the Simple Connectedness of Basins of Sinks for Iterations of Rational Maps, Preprint, Polish Academy of Sciences, Warsaw, 1987.
 [Sut]
 S. Sutherland, Finding Roots of Complex Polynomials with Newton's Method, (preprint) Institute for Mathematical Sciences, SUNY Stony Brook, 1990.
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Additional Information
Mako E. Haruta
Affiliation:
Department of Mathematics, University of Hartford, West Hartford, Connecticut 06117
Email:
mharuta@hartford.edu
DOI:
http://dx.doi.org/10.1090/S0002994799019273
PII:
S 00029947(99)019273
Keywords:
Newton's method,
basin of attraction
Received by editor(s):
February 27, 1995
Received by editor(s) in revised form:
September 22, 1996
Published electronically:
February 15, 1999
Article copyright:
© Copyright 1999
American Mathematical Society
