Complex function theory, 1780–1900

2003 ◽  
pp. 213-259 ◽  
Keyword(s):  
Function Theory ◽  
Complex Function ◽  
Complex Function Theory

The Topological Nature of Two Noguchi Theorems on Sequences of Holomorphic Mappings Between Complex Spaces

1995 ◽  
Vol 47 (6) ◽  
pp. 1240-1252
Author(s):  
James E. Joseph ◽  
Myung H. Kwack
Keyword(s):  
Function Theory ◽  
Riemann Sphere ◽  
Complex Function ◽  
Holomorphic Extension ◽  
Holomorphic Mappings ◽  
Classical Theorem ◽  
Complex Function Theory ◽  
Topological Methods ◽  
Complex Spaces ◽  
Picard Theorem

AbstractLet C,D,D* be, respectively, the complex plane, {z ∈ C : |z| < 1}, and D — {0}. If P1(C) is the Riemann sphere, the Big Picard theorem states that if ƒ:D* → P1(C) is holomorphic and P1(C) → ƒ(D*) n a s more than two elements, then ƒ has a holomorphic extension . Under certain assumptions on M, A and X ⊂ Y, combined efforts of Kiernan, Kobayashi and Kwack extended the theorem to all holomorphic ƒ: M → A → X. Relying on these results, measure theoretic theorems of Lelong and Wirtinger, and other properties of complex spaces, Noguchi proved in this context that if ƒ: M → A → X and ƒn: M → A → X are holomorphic for each n and ƒn → ƒ, then . In this paper we show that all of these theorems may be significantly generalized and improved by purely topological methods. We also apply our results to present a topological generalization of a classical theorem of Vitali from one variable complex function theory.

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