Geodesic
Cayley's problem
Applications of Mathematics, Tome 35 (1990) no. 2, pp. 140-146
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/AM.1990.104395
Classification : 39A10, 58C30, 58F08, 65H05, 65Q05
Keywords: Newton method; difference equation; series expansion; fixed point; discrete dynamical system; Julia set; Cayley’s problem; recurrence relations; analytic solution 
@article{10_21136_AM_1990_104395,
     author = {Petek, Peter},
     title = {Cayley's problem},
     journal = {Applications of Mathematics},
     pages = {140--146},
     year = {1990},
     volume = {35},
     number = {2},
     doi = {10.21136/AM.1990.104395},
     mrnumber = {1042849},
     zbl = {0709.39001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1990.104395/}
}
TY  - JOUR
AU  - Petek, Peter
TI  - Cayley's problem
JO  - Applications of Mathematics
PY  - 1990
SP  - 140
EP  - 146
VL  - 35
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/AM.1990.104395/
DO  - 10.21136/AM.1990.104395
LA  - en
ID  - 10_21136_AM_1990_104395
ER  - 
%0 Journal Article
%A Petek, Peter
%T Cayley's problem
%J Applications of Mathematics
%D 1990
%P 140-146
%V 35
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/AM.1990.104395/
%R 10.21136/AM.1990.104395
%G en
%F 10_21136_AM_1990_104395
Petek, Peter. Cayley's problem. Applications of Mathematics, Tome 35 (1990) no. 2, pp. 140-146. doi: 10.21136/AM.1990.104395

[1] A. Cayley: The Newton-Fourier imaginary problem. Amer. J. Math. II, 97 (1879).

[2] P. Petek: A Nonconverging Newton Sequence. Math. Magazine 56, no. 1, 43 - 45 (1983). | DOI | MR | Zbl

[3] G. Julia: Sur l'iteration des fonctions rationnelles. Journal de Math. Pure et Appl. 8, 47-245 (1918).

[4] P. Fatou: Sur les equations fonctionelles. Bull. Soc. Math. France, 47: 161 - 271, 48: 33 - 94, 208-314 (1919). | DOI | MR

[5] H. O. Peitgen D. Saupe F. Haeseler: Cayley's Problem and Julia Sets. Math. Intelligencer 6: 11-20 (1984). | DOI | MR

[6] P. Blanchard: Complex Analytic Dynamics on the Riemann Sphere. Bull. Amer. Math. Soc. 11, 85-141 (1984). | DOI | MR | Zbl

[7] C. L. Siegel: Iteration of Analytic Functions. Annals of Mathematics 43, 607-612 (1942). | DOI | MR | Zbl

[8] H. O. Peitgen P. H. Richter: The Beauty of Fractals. Springer-Verlag, Berlin, Heidelberg 1986. | MR

Cité par Sources :