Picard's Great Theorem

Every nonconstant entire function attains every complex value with at most one exception (Henrici 1988, p. 216; Apostol 1997). Furthermore, every analytic function assumes every complex value, with possibly one exception, infinitely often in any neighborhood of an essential singularity.

See also

Analytic Function, Essential Singularity, Neighborhood, Picard's Existence Theorem, Picard's Little Theorem

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Apostol, T. M. "Application to Picard's Theorem." §2.9 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 43-44, 1997.Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: Springer-Verlag, 1997.Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros. New York: Wiley, 1988.Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, 1968.Krantz, S. G. "Picard's Great Theorem." §10.5.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 140, 1999.Narasimhan, R. and Nievergelt, Y. Complex Analysis in One Variable. Boston: Birkhäuser, 2001.Remmert, R. Funktionentheorie 1. Berlin: Springer-Verlag, 1992.Remmert, R. Funktionentheorie 2. Berlin: Springer-Verlag, 1992.Trott, M. "Elementary Transcendental Functions." The Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 166, 2004.

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Picard's Great Theorem

Cite this as:

Weisstein, Eric W. "Picard's Great Theorem." From MathWorld--A Wolfram Web Resource.

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