Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T09:57:37.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Iteration Near a Fixed Point

Published online by Cambridge University Press:  05 September 2017

Cedric A. B. Smith
Get access Rights & Permissions [Opens in a new window]

Abstract

An exposition is given of the properties of the iterates of complex functions near a fixed point, with explicit expressions for their power series in certain cases. The relevance to problems in genetics and statistics is pointed out.

Type
Part IV — Mathematical Methods in Probability and Statistics
Information
Journal of Applied Probability , Volume 12 , Issue S1 , 1975 , pp. 157 - 173
Copyright
Copyright © 1975 Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abel, N. H. (1881) Détermination d'une fonction au moyen d'une équation qui ne contient qu'une seule variable. Oeuvres Complètes 2, 36–39.Google Scholar
Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Babbage, C. (1815) An essay towards a calculus of functions. Phil. Trans. R. Soc. 105, 389–423.Google Scholar
Bennett, A. A. (1915a) The iteration of functions of one variable. Ann. of Math. 17, 23–60.Google Scholar
Bennett, A. A. (1915b) A case of iteration in several variables. Ann. of Math. 17, 188–196.Google Scholar
Boole, G. (1860) A Treatise on the Calculus of Finite Differences. Macmillan, London.Google Scholar
Bromwich, T. J. I'A. (1931) An Introduction to the Theory of Infinite Series. Macmillan, London.Google Scholar
Domb, C. and Fisher, M. E. (1956) On iterative processes and functional equations. Proc. Camb. Phil. Soc. 52, 652–662.Google Scholar
Fatou, P. (1919, 1920a, 1920b) Sur les équations fonctionelles, I, II, III. Bull. Soc. Math. France 47, 161–271; 48, 33–94; 48, 208–314.CrossRefGoogle Scholar
Grevy, A. (1894) Études sur les équations fonctionelles. Ann. Sci. École Norm. Sup. Série 3, 11, 249–323.Google Scholar
Haldane, J. B. S. (1932) On the non-linear functional equation ?xn = k? (xn). Proc. Camb. Phil. Soc. 28, 234–243.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.Google Scholar
Julia, G. (1924) Sur quelques applications de la représentation conforme à la résolution d'équations fonctionelles. Bull. Soc. Math. France 52, 279–315.Google Scholar
Koenigs, G. (1884) Recherches sur les intégrales de certaines équations fonctionelles. Ann. Sci. École Norm. Sup. Série 3, 1, Supplément S1–S45.Google Scholar
Koenigs, G. (1885) Nouvelles recherches sur les équations fonctionelles. Ann. Sci. École Norm. Sup. Série 3, 2, 385–404.CrossRefGoogle Scholar
Leau, L. (1897) Études sur les équations fonctionelles à une ou a plusieurs variables. Ann. Fac. Sci. Univ. Toulouse 11, E1–E110.CrossRefGoogle Scholar
Lemeray, E. M. (1898) Sur quelques algorithmes généraux et sur l'itération. Bull. Soc. Math. France 27, 10–15.Google Scholar
Menger, K. (1953) Calculus: A Modern Approach. Illinois Institute of Technology, Chicago.Google Scholar
Picard, E. (1928) Leçons sur quelques Équations Fonctionelles. Gauthier-Villars, Paris.Google Scholar
Rausenberger, O. (1881) Theorie der allgemeinen Periodicität. Math. Ann. 18, 379–409.Google Scholar
Schroder, E. (1870) Über iterierte Functionen. Math. Ann. 3, 296–322.Google Scholar
Siegel, C. L. (1942) Iterations of Analytic Functions. Ann. of Math. 43, 607–612.Google Scholar
Smith, C. A. B. (1942) On Tests of Significance. Ph. D. Thesis, University of Cambridge.Google Scholar
Szekeres, G. (1958) Regular iteration of real and complex functions. Acta Math. 100, 203–258.Google Scholar
Turnbull, H. W. (1934) Power vectors. Proc. London Math. Soc. (2) 37, 106–146.Google Scholar
Walker, A. G. (1946) Commutative functions (I and II). Quart. J. Math. Oxford Ser. 17, 65–92.CrossRefGoogle Scholar
Walker, A. G. and Batty, J. S. (1946) Non-integral function powers. Quart. J. Math. Oxford Ser. 17, 145–152.Google Scholar
Watson, H. W. and Galton, F. (1874) On the probability of the extinction of families. J. Anthropol. Inst. Great Britain and Ireland 4, 138–144.Google Scholar