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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
MathSciNet review:
637705
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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.
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Douglas
S. Bridges, On the isolation of zeroes of an analytic
function, Pacific J. Math. 96 (1981), no. 1,
13–22. MR
634759 (82k:03095)
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B. Conway, Functions of one complex variable, Springer-Verlag,
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Dieudonné, Foundations of modern analysis, Pure and
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E. C. Titchmarsh, The theory of functions, 2nd ed. (corrected), Clarendon Press, Oxford, 1968.
- [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.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9947-1982-0637705-7
PII:
S 0002-9947(1982)0637705-7
Article copyright:
© Copyright 1982 American Mathematical Society
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