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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Picard’s theorem
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by Douglas Bridges, Allan Calder, William Julian, Ray Mines and Fred Richman PDF
Trans. Amer. Math. Soc. 269 (1982), 513-520 Request permission

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
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  • A. S. B. Holland, Introduction to the theory of entire functions, Pure and Applied Mathematics, Vol. 56, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0447572
  • Paul Montel, Sur les familles de fonctions analytiques qui admettent des valeurs exceptionnelles dans un domaine, Ann. Sci. École Norm. Sup. (3) 29 (1912), 487–535 (French). MR 1509154
  • E. Picard, Sur les fonctions analytiques uniformes dans le voisinage d’un point singulier essentiel, C. R. Acad. Sci. 89 (1879), 745-747. F. Schottky, Über den Picard’schen Satz und die Borel’schen Ungleichungen, Sitzung. der K. Preussischen Akad. der Wissenschaften, Berlin, 1904, pp. 1244-1262. E. C. Titchmarsh, The theory of functions, 2nd ed. (corrected), Clarendon Press, Oxford, 1968.
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • 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