Picard’s Theorem

  • Raghavan Narasimhan

Abstract

In this chapter, we shall prove the so-called “big” theorem of Picard which asserts that a holomorphic function with an (isolated) essential singularity assumes every value with at most one exception in any neighborhood of that singularity.

Keywords

Compact Subset Essential Singularity Compact Complex Manifold Nevanlinna Theory Schwarz Lemma 
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References : Chapter 4

  1. [1]
    Ahlfors, L.V.: An extension of Schwarz’s lemma. Trans. Amer. Math. Soc. 43 (1938), 359–364.MathSciNetGoogle Scholar
  2. [2]
    Bloch, A.: Les théorèmes de M. Valiron sur les fonctions entières, et la théorie de l’uniformisation. Ann. Fac. des Sciences, Univ. de Toulous. 17 (1925), 1–22.MathSciNetCrossRefGoogle Scholar
  3. [2]
    Bloch, A.: See also a short version, with the same title, in C. R. Acad. Sci. Pari. 178 (1924), 2051–2052.MATHGoogle Scholar
  4. [3]
    Borel, E.: Sur les zéros des fonctions entières. Acta Math. 20 (1897), 357–396.MathSciNetMATHCrossRefGoogle Scholar
  5. [4]
    Conway, J. B. : Functions of one complex variable. Springer, 1973.MATHCrossRefGoogle Scholar
  6. [5]
    Copson, E. T. : An introduction to the theory of functions of a complex variabl., Oxford University Press, 1935 (and later reprints).Google Scholar
  7. [6]
    Grauert, H. and H. Reckziegel : Hermitische Metriken und normale Familien holomorpher Abbildungen. Math. Zeit. 89 (1965), 108–125.MathSciNetMATHCrossRefGoogle Scholar
  8. [7]
    Griffiths, P. A. : Entire holomorphic mappings in one and several variables. Annals of Math. Studies, Princeton, 1976.MATHGoogle Scholar
  9. [8]
    Heins, M.: Complex functions theory. New York: Academic Press, 1968.Google Scholar
  10. [9]
    Hurwitz, A. and R. Courant : Funktionentheorie. 4th ed. with an appendix by H. Röhrl, Springer, 1964.MATHGoogle Scholar
  11. [10]
    Kodaira, K.: Holomorphic mappings of polydiscs into compact complex manifolds. J. Diff. Geometr. 6 (1971), 33–46.MathSciNetMATHGoogle Scholar
  12. [11]
    Landau, E.: Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheori., 2nd ed. Springer, 1929 (Chelsea reprint, 1946).MATHGoogle Scholar
  13. [12]
    Montel, P.: Leçons sur les familles normales de fonctions analytiques. Paris, 1927.MATHGoogle Scholar
  14. [13]
    Nevanlinna, R.: Le théorème de Picard-Borel et la théorie des functions méromorphes. Paris, 1929.Google Scholar
  15. [14]
    Nevanlinna, R.: Eindeutige analytische Funktionen. Springer, 1936 (English translation : Analytic Function., Springer, 1970.)Google Scholar
  16. [15]
    Valiron, G.: Sur les théorèmes de Mm. Bloch, Landau, Montel et Schottky. C. R. Acad. Sci. Pari., 183 (1926), 728–730.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Raghavan Narasimhan
    • 1
  1. 1.Department of MathematicsThe University of ChicagoChicagoUSA

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