Cayley's problem
Aplikace matematiky (1990)
- Volume: 35, Issue: 2, page 140-146
- ISSN: 0862-7940
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topPetek, Peter. "Cayley's problem." Aplikace matematiky 35.2 (1990): 140-146. <http://eudml.org/doc/15617>.
@article{Petek1990,
abstract = {Newton's method for computation of a square root yields a difference equation which can be solved using the hyperbolic cotangent function. For the computation of the third root Newton's sequence presents a harder problem, which already Cayley was trying to solve. In the present paper two mutually inverse functions are defined in order to solve the difference equation, instead of the hyperbolic cotangent and its inverse. Several coefficients in the expansion around the fixed points are obtained, and the expansions are glued together in the region of overlapping.},
author = {Petek, Peter},
journal = {Aplikace matematiky},
keywords = {Newton method; difference equation; series expansion; fixed point; discrete dynamical system; Julia set; Cayley’s problem; recurrence relations; analytic solution; Cayley's problem; recurrence relations; analytic solution; difference equation; Newton's method},
language = {eng},
number = {2},
pages = {140-146},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Cayley's problem},
url = {http://eudml.org/doc/15617},
volume = {35},
year = {1990},
}
TY - JOUR
AU - Petek, Peter
TI - Cayley's problem
JO - Aplikace matematiky
PY - 1990
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 35
IS - 2
SP - 140
EP - 146
AB - Newton's method for computation of a square root yields a difference equation which can be solved using the hyperbolic cotangent function. For the computation of the third root Newton's sequence presents a harder problem, which already Cayley was trying to solve. In the present paper two mutually inverse functions are defined in order to solve the difference equation, instead of the hyperbolic cotangent and its inverse. Several coefficients in the expansion around the fixed points are obtained, and the expansions are glued together in the region of overlapping.
LA - eng
KW - Newton method; difference equation; series expansion; fixed point; discrete dynamical system; Julia set; Cayley’s problem; recurrence relations; analytic solution; Cayley's problem; recurrence relations; analytic solution; difference equation; Newton's method
UR - http://eudml.org/doc/15617
ER -
References
top- A. Cayley, The Newton-Fourier imaginary problem, Amer. J. Math. II, 97 (1879).
- P. Petek, 10.1080/0025570X.1983.11977016, Math. Magazine 56, no. 1, 43 - 45 (1983). (1983) Zbl0505.10006MR0692174DOI10.1080/0025570X.1983.11977016
- G. Julia, Sur l'iteration des fonctions rationnelles, Journal de Math. Pure et Appl. 8, 47-245 (1918). (1918)
- P. Fatou, 10.24033/bsmf.998, Bull. Soc. Math. France, 47: 161 - 271, 48: 33 - 94, 208-314 (1919). (1919) MR1504787DOI10.24033/bsmf.998
- H. O. Peitgen D. Saupe F. Haeseler, 10.1007/BF03024150, Math. Intelligencer 6: 11-20 (1984). (1984) MR0738904DOI10.1007/BF03024150
- P. Blanchard, 10.1090/S0273-0979-1984-15240-6, Bull. Amer. Math. Soc. 11, 85-141 (1984). (1984) Zbl0558.58017MR0741725DOI10.1090/S0273-0979-1984-15240-6
- C. L. Siegel, 10.2307/1968952, Annals of Mathematics 43, 607-612 (1942). (1942) Zbl0061.14904MR0007044DOI10.2307/1968952
- H. O. Peitgen P. H. Richter, The Beauty of Fractals, Springer-Verlag, Berlin, Heidelberg 1986. (1986) MR0852695
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