On the distortion of schlicht non-conformal mappings and on a related extension of Picard’s theorem

2020 ◽  
pp. 371-374
Author(s):  
Herbert Grötzsch
Keyword(s):  
Conformal Mappings ◽  
Picard’S Theorem ◽  
Picard's Theorem

Picard’s Theorem

2001 ◽  
pp. 87-96
Author(s):  
Raghavan Narasimhan ◽  
Yves Nievergelt
Keyword(s):  
Picard’S Theorem ◽  
Picard's Theorem

Picard's Theorem Without Tears

1978 ◽  
Vol 85 (4) ◽  
pp. 265-268
Author(s):  
Lawrence Zalcman
Keyword(s):  
Picard’S Theorem ◽  
Picard's Theorem

An Equivalent Form of Picard’s Theorem and Beyond

2018 ◽  
Vol 61 (1) ◽  
pp. 142-148 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Functional Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Equivalent Form ◽  
Entire Solutions ◽  
Partial Differential ◽  
Picard’S Theorem ◽  
Picard's Theorem

AbstractThis paper gives an equivalent form of Picard’s theorem via entire solutions of the functional equation f2 + g2 = 1 and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

scholarly journals Picard’s theorem

1982 ◽  
Vol 269 (2) ◽  
pp. 513-513
Author(s):  
Douglas Bridges ◽  
Allan Calder ◽  
William Julian ◽  
Ray Mines ◽  
Fred Richman
Keyword(s):  
Picard’S Theorem ◽  
Picard's Theorem

Picard's Theorem

1972 ◽  
Vol 79 (9) ◽  
pp. 1020
Author(s):  
James Fabrey
Keyword(s):  
Picard’S Theorem ◽  
Picard's Theorem

The Algebraic Independence of Certain Exponential Functions

1976 ◽  
Vol 28 (5) ◽  
pp. 968-976
Author(s):  
W. Dale Brownawell
Keyword(s):  
Meromorphic Functions ◽  
Entire Functions ◽  
General Theorem ◽  
Algebraic Independence ◽  
Exponential Functions ◽  
Linearly Independent ◽  
Borel’S Theorem ◽  
Picard’S Theorem ◽  
Picard's Theorem ◽  
Special Case

In 1897 E. Borel proved a general theorem which implied as a special case the following result equivalent to his celebrated generalization of Picard's theorem [2]: If f1 … ,fm are entire functions such that for each, C then the functions exp f1, … , exp f/m are linearly independent over C. In 1929 R. Nevanlinna [6] extended Borel's theorem to consider arbitrary C-linearly independent meromorphic functions < ϕi, … , < ϕm satisfying < ϕ1 + … + ϕm = 1.

scholarly journals Algebraic degeneracy theorem for holomorphic mappings into smooth projective algebraic varieties

1981 ◽  
Vol 84 ◽  
pp. 209-218
Author(s):  
Yoshihiro Aihara ◽  
Seiki Mori
Keyword(s):  
General Position ◽  
Holomorphic Mapping ◽  
Holomorphic Mappings ◽  
Holomorphic Curve ◽  
Algebraic Varieties ◽  
Picard Theorem ◽  
Picard’S Theorem ◽  
Picard's Theorem ◽  
Algebraic Degeneracy

The famous Picard theorem states that a holomorphic mapping f: C → P1(C) omitting distinct three points must be constant. Borel [1] showed that a non-degenerate holomorphic curve can miss at most n + 1 hyperplanes in Pn(C) in general position, thus extending Picard’s theorem (n = 1). Recently, Fujimoto [3], Green [4] and [5] obtained many Picard type theorems using Borel’s methods for holomorphic mappings.

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