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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References : Chapter 4
Ahlfors, L.V.: An extension of Schwarz’s lemma. Trans. Amer. Math. Soc. 43 (1938), 359–364.
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.
Bloch, A.: See also a short version, with the same title, in C. R. Acad. Sci. Pari. 178 (1924), 2051–2052.
Borel, E.: Sur les zéros des fonctions entières. Acta Math. 20 (1897), 357–396.
Conway, J. B. : Functions of one complex variable. Springer, 1973.
Copson, E. T. : An introduction to the theory of functions of a complex variabl., Oxford University Press, 1935 (and later reprints).
Grauert, H. and H. Reckziegel : Hermitische Metriken und normale Familien holomorpher Abbildungen. Math. Zeit. 89 (1965), 108–125.
Griffiths, P. A. : Entire holomorphic mappings in one and several variables. Annals of Math. Studies, Princeton, 1976.
Heins, M.: Complex functions theory. New York: Academic Press, 1968.
Hurwitz, A. and R. Courant : Funktionentheorie. 4th ed. with an appendix by H. Röhrl, Springer, 1964.
Kodaira, K.: Holomorphic mappings of polydiscs into compact complex manifolds. J. Diff. Geometr. 6 (1971), 33–46.
Landau, E.: Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheori., 2nd ed. Springer, 1929 (Chelsea reprint, 1946).
Montel, P.: Leçons sur les familles normales de fonctions analytiques. Paris, 1927.
Nevanlinna, R.: Le théorème de Picard-Borel et la théorie des functions méromorphes. Paris, 1929.
Nevanlinna, R.: Eindeutige analytische Funktionen. Springer, 1936 (English translation : Analytic Function., Springer, 1970.)
Valiron, G.: Sur les théorèmes de Mm. Bloch, Landau, Montel et Schottky. C. R. Acad. Sci. Pari., 183 (1926), 728–730.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1985 Springer Science+Business Media New York
About this chapter
Cite this chapter
Narasimhan, R. (1985). Picard’s Theorem. In: Complex Analysis in one Variable. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-1106-6_4
Download citation
DOI: https://doi.org/10.1007/978-1-4757-1106-6_4
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-3237-3
Online ISBN: 978-1-4757-1106-6
eBook Packages: Springer Book Archive