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AuthorSario, Leo. author
TitleValue Distribution Theory [electronic resource] / by Leo Sario, Kiyoshi Noshiro
ImprintNew York, NY : Springer New York : Imprint: Springer, 1966
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Descript XI, 236 p. online resource


The purpose of this research monograph is to build up a modern value distribution theory for complex analytic mappings between abstract Riemann surfaces. All results presented herein are new in that, apart from the classical background material in the last chapter, there is no overยญ lapping with any existing monograph on merom orphic functions. Broadly speaking the division of the book is as follows: The Introduction and Chapters I to III deal mainly with the theory of mappings of arbitrary Riemann surfaces as developed by the first named author; Chapter IV, due to Nakai, is devoted to meromorphic functions on parabolic surfaces; Chapter V contains Matsumoto's results on Picard sets; Chapter VI, preยญ dominantly due to the second named author, presents the so-called nonintegrated forms of the main theorems and includes some joint work by both authors. For a complete list of writers whose results have been discussed we refer to the Author Index


1. Historical -- 2. New metric -- 3. The fundamental A-, B-, and C-functions -- 4. Method of areal proximity -- 5. Summary -- I Mappings Into Closed Riemann Surfaces -- ยง1. Mappings of Arbitrary Riemann Surfaces -- ยง2. Meromorphic Functions on Arbitrary Riemann Surfaces -- ยง3. Surfaces RS and Conformal Metrics -- II Mappings Into Open Riemann Surfaces -- ยง1. Principal Functions -- ยง2. Proximity Functions on Arbitrary Riemann Surfaces -- ยง3. Analytic Mappings -- III Functions of Bounded Characteristic -- ยง1. Decomposition -- ยง2. The Class OMB -- IV Functions on Parabolic Riemann Surfaces -- ยง1. The Evans-Selberg Potential -- ยง2. Meromorphic Functions in a Boundary Neighborhood -- V Picard Sets -- ยง1. Infinite Picard Sets -- ยง2. Finite Picard Sets -- VI Riemannian Images -- ยง1. Mean Sheet Numbers -- ยง2. Euler Characteristic -- ยง3. Islands and Peninsulas -- ยง4. Meromorphic Functions -- ยง5. Mappings of Arbitrary Riemann Surfaces -- Appendix I. Basic Properties of Riemann Surfaces -- Appendix II. Gaussian Mapping of Arbitrary Minimal Surfaces -- 1. Triple connectivity -- 2. Arbitrary connectivity -- 3. Arbitrary genus -- 4. Arbitrary genus and connectivity -- 5. Gaussian mapping -- 6. Picard directions -- 7. Islands and peninsulas -- 8. Regular exhaustions -- 9. Open questions -- Author Index

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