Value Distribution Theory, Softcover reprint of the original 1st ed. 1966 The university series in higher mathematics Series

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 over­ 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, pre­ 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.- §1. Mappings of Arbitrary Riemann Surfaces.- §2. Meromorphic Functions on Arbitrary Riemann Surfaces.- §3. Surfaces RS and Conformal Metrics.- II Mappings Into Open Riemann Surfaces.- §1. Principal Functions.- §2. Proximity Functions on Arbitrary Riemann Surfaces.- §3. Analytic Mappings.- III Functions of Bounded Characteristic.- §1. Decomposition.- §2. The Class OMB.- IV Functions on Parabolic Riemann Surfaces.- §1. The Evans-Selberg Potential.- §2. Meromorphic Functions in a Boundary Neighborhood.- V Picard Sets.- §1. Infinite Picard Sets.- §2. Finite Picard Sets.- VI Riemannian Images.- §1. Mean Sheet Numbers.- §2. Euler Characteristic.- §3. Islands and Peninsulas.- §4. Meromorphic Functions.- §5. 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.