Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Picard's theorem


Authors: Douglas Bridges, Allan Calder, William Julian, Ray Mines and Fred Richman
Journal: Trans. Amer. Math. Soc. 269 (1982), 513-520
MSC: Primary 03F65; Secondary 30B10, 30D35
DOI: https://doi.org/10.1090/S0002-9947-1982-0637705-7
MathSciNet review: 637705
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with the numerical content of Picard's Theorem. Two classically equivalent versions of this theorem are proved which are distinct from a computational point of view. The proofs are elementary, and constructive in the sense of Bishop. A Brouwerian counterexample is given to the original version of the theorem.


References [Enhancements On Off] (What's this?)

  • [1] Errett Bishop, Foundations of constructive analysis, McGraw-Hill, New York, 1967. MR 0221878 (36:4930)
  • [2] D. S. Bridges, On the isolation of zeroes of an analytic function, Pacific J. Math. (to appear). MR 634759 (82k:03095)
  • [3] J. B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Math., no. 11, Springer-Verlag, Berlin, 1975. MR 0447532 (56:5843)
  • [4] J. Dieudonné, Foundations of modern analysis, Academic Press, New York, 1960. MR 0120319 (22:11074)
  • [5] W. H. J. Fuchs, Topics in the theory of functions of one complex variable, Van Nostrand, Princeton, N. J., 1967. MR 0220902 (36:3954)
  • [6] A. S. B. Holland, Introduction to the theory of entire functions, Pure and Appl. Math., vol. 56, Academic Press, New York, 1973. MR 0447572 (56:5882)
  • [7] P. Montel, Sur les familles de fonctions analytiques qui admettent des valeurs exceptionnelles dans un domaine, Ann. Ecole Norm. Sup. (3) 23 (1912), 487-535. MR 1509154
  • [8] E. Picard, Sur les fonctions analytiques uniformes dans le voisinage d'un point singulier essentiel, C. R. Acad. Sci. 89 (1879), 745-747.
  • [9] F. Schottky, Über den Picard'schen Satz und die Borel'schen Ungleichungen, Sitzung. der K. Preussischen Akad. der Wissenschaften, Berlin, 1904, pp. 1244-1262.
  • [10] E. C. Titchmarsh, The theory of functions, 2nd ed. (corrected), Clarendon Press, Oxford, 1968.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03F65, 30B10, 30D35

Retrieve articles in all journals with MSC: 03F65, 30B10, 30D35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0637705-7
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society