| Title:
|
Cayley's problem (English) |
| Author:
|
Petek, Peter |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
35 |
| Issue:
|
2 |
| Year:
|
1990 |
| Pages:
|
140-146 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Newton method |
| Keyword:
|
difference equation |
| Keyword:
|
series expansion |
| Keyword:
|
fixed point |
| Keyword:
|
discrete dynamical system |
| Keyword:
|
Julia set |
| Keyword:
|
Cayley’s problem |
| Keyword:
|
recurrence relations |
| Keyword:
|
analytic solution |
| MSC:
|
39A10 |
| MSC:
|
58C30 |
| MSC:
|
58F08 |
| MSC:
|
65H05 |
| MSC:
|
65Q05 |
| idZBL:
|
Zbl 0709.39001 |
| idMR:
|
MR1042849 |
| DOI:
|
10.21136/AM.1990.104395 |
| . |
| Date available:
|
2008-05-20T18:38:47Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104395 |
| . |
| Reference:
|
[1] A. Cayley: The Newton-Fourier imaginary problem.Amer. J. Math. II, 97 (1879). |
| Reference:
|
[2] P. Petek: A Nonconverging Newton Sequence.Math. Magazine 56, no. 1, 43 - 45 (1983). Zbl 0505.10006, MR 0692174, 10.1080/0025570X.1983.11977016 |
| Reference:
|
[3] G. Julia: Sur l'iteration des fonctions rationnelles.Journal de Math. Pure et Appl. 8, 47-245 (1918). |
| Reference:
|
[4] P. Fatou: Sur les equations fonctionelles.Bull. Soc. Math. France, 47: 161 - 271, 48: 33 - 94, 208-314 (1919). MR 1504787, 10.24033/bsmf.998 |
| Reference:
|
[5] H. O. Peitgen D. Saupe F. Haeseler: Cayley's Problem and Julia Sets.Math. Intelligencer 6: 11-20 (1984). MR 0738904, 10.1007/BF03024150 |
| Reference:
|
[6] P. Blanchard: Complex Analytic Dynamics on the Riemann Sphere.Bull. Amer. Math. Soc. 11, 85-141 (1984). Zbl 0558.58017, MR 0741725, 10.1090/S0273-0979-1984-15240-6 |
| Reference:
|
[7] C. L. Siegel: Iteration of Analytic Functions.Annals of Mathematics 43, 607-612 (1942). Zbl 0061.14904, MR 0007044, 10.2307/1968952 |
| Reference:
|
[8] H. O. Peitgen P. H. Richter: The Beauty of Fractals.Springer-Verlag, Berlin, Heidelberg 1986. MR 0852695 |
| . |
