Newton's method on

the complex exponential function

Author:
Mako E. Haruta

Journal:
Trans. Amer. Math. Soc. **351** (1999), 2499-2513

MSC (1991):
Primary 58F23

Published electronically:
February 15, 1999

MathSciNet review:
1422898

Full-text 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.

**[A]**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, Rheinisch-Westfälische 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 Newton-Fourier Method to an Imaginary Root of an Equation*,*Quart. J. of Pure and Applied Math.***XVI**(1879), 179-185.**[CGS]**James H. Curry, Lucy Garnett, and Dennis Sullivan,*On the iteration of a rational function: computer experiments with Newton’s method*, Comm. Math. Phys.**91**(1983), no. 2, 267–277. MR**723551****[H]**Mike Hurley,*Multiple attractors in Newton’s method*, Ergodic Theory Dynam. Systems**6**(1986), no. 4, 561–569. MR**873432**, 10.1017/S0143385700003692**[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), no. 2, 238–252. MR**731865**, 10.1137/0515020**[HP]**B. A. Kupershmidt,*Canonical property of the Miura maps between the mKP and KP hierarchies, continuous and discrete*, Comm. Math. Phys.**167**(1995), no. 2, 351–371. MR**1316510****[Kr]**H. Kreite,*A Newton's Method Case with a Basin of Infinite Area*, Preprint, Fakultät und Institut für Mathematik, Ruhr-Universität 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]**Curt McMullen,*Families of rational maps and iterative root-finding algorithms*, Ann. of Math. (2)**125**(1987), no. 3, 467–493. MR**890160**, 10.2307/1971408**[MSS]**R. Mañé, P. Sad, and D. Sullivan,*On the dynamics of rational maps*, Ann. Sci. École Norm. Sup. (4)**16**(1983), no. 2, 193–217. MR**732343****[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:
https://doi.org/10.1090/S0002-9947-99-01927-3

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