亞純函數(shù)值分布理論

前言

This book is devoted to the study of value distribution of functions which are mero-morphic on the complex plane or in an angular domain with vertex at the origin. Wecharacterize such meromorphic functions in terms of distribution of some of theirvalue points. The study, together with certain related topics, is known as theory ofvalue distribution of meromorphic functions. The theory is too vast to be justifiedwithin a single work. Therefore we selected and organized the content based on theirsignificant importance to our understanding and interests in this book. I gladly ac-knowledge my indebtedness in particular to the books of M. Tsuji, A. A. Goldbergand I. V. Ostrovskii, Yang L. and the papers of A. Eremenko.   An outline of the book is provided below. The introduction of the Nevanlinnacharacteristic to the study of meromorphic functions is a new starting symbol ofthe theory of value distribution. The Nevanlinna characteristic is powerful, and itsthought has been used to produce various characteristics such as the Nevanlinnacharacteristic and Tsuji characteristic for an angular domain. And from geometricpoint of view, namely the Ahlfors theory of covering surfaces, the Ahlfors-Shimizucharacteristic have also been introduced. These characteristics are real-valued func-tions defined on the positive real axis. Therefore, in the first chapter, we collect thebasic results about positive real functions that are often used in the study of mero-morphic function theory. Some of these results are distributed in other books, somein published papers, and some have been newly established in order to serve ourspecific objectives in the book.

內(nèi)容概要

本書共7章，研究在復(fù)平面上或在以原點為頂點的角域上亞純的函數(shù)的值分布，即通過某些值點來刻畫亞純函數(shù)。前兩章研究各類特征函數(shù)及這樣的實函數(shù)的性質(zhì)。第3、4章放在新引入的奇異方向——T方向，包括存在性、分布，與其他方向的關(guān)系上，T方向與分布值和虧值總數(shù)的關(guān)系。射線分布值確定亞純函數(shù)的增長性的問題在第5章詳細研究。第6章研究亞純函數(shù)對應(yīng)的Riemann曲面，逆函數(shù)的奇異性及其與不動點的關(guān)系。最后一章介紹具有重要地位的ENevanlinna猜想的Eremenko應(yīng)用位勢論的證明。

作者簡介

鄭建華，Dr. Jianhua Zheng is a Professor at the Department of MathematicalSciences, Tsinghua University, China.

書籍目錄

1 Preliminaries of Real Functions   1.1 Functions of a Real Variable   1.1.1 The Order and Lower Order of a Real Function   1.1.2 The P61ya Peak Sequence of a Real Function   1.1.3 The Regularity of a Real Function   1.1.4 Quasi-invariance of Inequalities   1.2 Integral Formula and Integral Inequalities   1.2.1 The Green Formula for Functions with Two Real Variables   1.2.2 Several Integral Inequalities   References 2 Characteristics of a Meromorphic Function   2.1 Nevanlinna's Characteristic in a Domain   2.2 Nevanlinna's Characteristic in an Angle   2.3 Tsuji's Characteristic   2.4 Ahlfors-Shimizu's Characteristic   2.5 Estimates of the Error Terms   2.6 Characteristic of Derivative of a Meromorphic Function   2.7 Meromorphic Functions in an Angular Domain   2.8 Deficiency and Deficient Values   2.9 Uniqueness of Meromorphic Functions Related to Some Angular Domains   References 3 T Directions of a Meromorphic Function   3.1 Notation and Existence of T Directions   3.2 T Directions Dealing with Small Functions   3.3 Connection Among T Directions and Other Directions   3.4 Singular Directions Dealing with Derivatives   3.5 The Common T Directions of a Meromorphic Function and Its Derivatives   3.6 Distribution of the Julia, Borel Directions and T Directions   3.7 Singular Directions of Meromorphic Solutions of Some Equations   3.8 Value Distribution of Algebroid Functions   References 4 Argument Distribution and Deficient Values   4.1 Deficient Values and T Directions   4.2 Retrospection   References 5 Meromorphic Functions with Radially Distributed Values   5.1 Growth of Such Meromorphic Functions   5.2 Growth of Such Meromorphic Functions with Finite Lower Order   5.3 Retrospection   References 6 Singular Values of Meromorphic Functions   6.1 Riemann Surfaces and Singularities   6.2 Density of Singularities   6.3 Meromorphic Functions of Bounded Type   References 7 The Potential Theory in Value Distribution   7.1 Signed Measure and Distributions   7.2 8-Subharmonic Functions   7.2.1 Basic Results Concerning 8-Subharmonic Functions   7.2.2 Normality of Family of 8-Subharmonic Functions   7.2.3 The Nevanlinna Theory of 8-Subharmonic Functions   7.3 Eremenko's Proof of the Nevanlinna Conjecture   References   Index

章節(jié)摘錄

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《亞純函數(shù)值分布理論》：Value Distribution of Meromorphic Functions focuses on functionsmeromorphic in an angle or on the complex plane, T directions, deficientvalues, singular values, potential theory in value distribution and theproof of the celebrated Nevanlinna conjecture. The book introducesvarious characteristics of meromorphic functions and their connections,several aspects of new singular directions, new results on estimates of thenumber of deficient values, new results on singular values and behavioursof subharmonic functions which are the foundation for further discussionon the proof of the Nevanlinna conjecture. The independent significanceof normality of subharmonic function family is emphasized. This book isdesigned for scientists, engineers and post graduated students engaged inComplex Analysis and Meromorphic Functions.

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