復分析I

前言

　　要使我國的數(shù)學事業(yè)更好地發(fā)展起來，需要數(shù)學家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學家創(chuàng)造更有利的發(fā)展數(shù)學事業(yè)的外部環(huán)境，這主要是加強對數(shù)學事業(yè)的支持與投資力度，使數(shù)學家有較好的工作與生活條件，其中也包括改善與加強數(shù)學的出版工作?！　某霭娣矫鎭碇v，除了較好較快地出版我們自己的成果外，引進國外的先進出版物無疑也是十分重要與必不可少的。從數(shù)學來說，施普林格（springer）出版社至今仍然是世界上最具權威的出版社?？茖W出版社影印一批他們出版的好的新書，使我國廣大數(shù)學家能以較低的價格購買，特別是在邊遠地區(qū)工作的數(shù)學家能普遍見到這些書，無疑是對推動我國數(shù)學的科研與教學十分有益的事?！　∵@次科學出版社購買了版權，一次影印了23本施普林格出版社出版的數(shù)學書，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書中，包括基礎數(shù)學書5本，應用數(shù)學書6本與計算數(shù)學書12本，其中有些書也具有交叉性質。這些書都是很新的，2000年以后出版的占絕大部分，共計16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數(shù)學某方面的前沿，例如基礎數(shù)學中的數(shù)論、代數(shù)與拓撲三本，都是由該領域大數(shù)學家編著的“數(shù)學百科全書”的分冊。對從事這方面研究的數(shù)學家了解該領域的前沿與全貌很有幫助。按照學科的特點，基礎數(shù)學類的書以“經典”為主，應用和計算數(shù)學類的書以“前沿”為主。這些書的作者多數(shù)是國際知名的大數(shù)學家，例如《拓撲學》一書的作者諾維科夫是俄羅斯科學院的院士，曾獲“菲爾茲獎”和“沃爾夫數(shù)學獎”。這些大數(shù)學家的著作無疑將會對我國的科研人員起到非常好的指導作用。　　當然，23本書只能涵蓋數(shù)學的一部分，所以，這項工作還應該繼續(xù)做下去。更進一步，有些讀者面較廣的好書還應該翻譯成中文出版，使之有更大的讀者群?！　】傊?，我對科學出版社影印施普林格出版社的部分數(shù)學著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績。

內容概要

The first part of the volume contains a comprehensive description of the theory of entire and meromorphic functions of one complex variable and its applications. It includes the fundamental notions,methods and results on the growth of entire functions and the distribution of their zeros, the RolfNevanlinna theory of distribution of values of meromorphic functions including the inverse problem,the theory of completely regular growth, the concept of limit sets for entire and subharmonic functions. The authors describe the applications to the interpolation by entire functions, to entire and meromorphic solutions of ordinary differential equations, to the Riemann boundary problem with an infinite index and to the arithmetic of the convolution semigroup of probability distributions.    Polyanalytic functions form one of the most natural generalizations of analytic functions and are described in Part II. They emerged for the first time in plane elasticity theory where they found important applications(due to Kolossof, Mushelishvili etc.). This contribution contains a detailed review of recent investigations concerning the function-theoretical pecularities ofpolyanalytic functions(boundarybehaviour, value distributions, degeneration, uniqueness etc.).Polyanalytic functions have many points of contact with such fields of analysis as polyharmonic functions, Nevanlinna Theory,meromorphic curves, cluster set theory, functions of several complex variables etc.

書籍目錄

COntentsIntroduct.ionChapter l.General Theorems on the Asymptotic Behavior of Entire and Meromorphic Functions(A.AGol’dberg，BYa.Levin.V.Ostrovskii)  §1.Characteristics of Asymptotic Behavior    §2.Relation Between Growth and Decrease  §3.Relation Between the Indicator of sD Entire Function andSingularities of Its Borel Transform  §4.Wiman-vallrinTheory  Chapter 2.The Connection Between the Growth of an EntireFunction and the Distribution of Its Zeros(B.Ya.Levin.V.Ostrovskii)  §1.Classical Results   §2.Entire Functions of Completely Regular Growth  §3.Entire Functions of Exponential Type with Restrictions on theReal Axis.  §4.Exceptional Sets    §5.Two-Tcrm Asyrnptotics A.A.Gol，dberg，B.Ya.LevinI.V.Ostrovskfi  §6.Approximation of a Subharmonic Function by the Logarithm of the Modulus of an Entire Function    §7.The Relation Between the Growth and Distribution of Zeros and Fourier Coefl~cientsChapter 3.Limit Sets of Entire and Subharmonic Functions(VS.Azarin).  §1.Principal Notations and Theorems    §2.Limit Sets and Their Dcation to Other Characteristics    §3.Applications of Limit Sets  §4.Limit Sets ss Dynamical Systems  Chapter 4Interpolation by Entire Functions(B.YaLevin，V.A.Tkachenko).  §1.Newton’S Interpolation Series  §2.Abel-conteharoff Interpolation Series  §3.Gelfond，s Moments Problem  §4.Lagrange’8 Interpolation Series    §5.Interpolation Techniques Based on Solving the良Problem    §6.The Lagrange Interpolation Process in Some Normed Spaces  Chapter 5.Distribution of vahues of Meromorphic Functions(AA.Gol，dberg)    §1.Main Nevanlinna Theorems.Nevanlinna Deficient VbLlUes and Deficient Functions  §2.Inverse Problems of Value DistributionTheory  §3.The Ahlfors了heory    §4.valironDeficiencies  §5.Exceptional Values in the Sense of Petrenk0    §6.Asymptotic Curves and Asymptotic Values.  §7.Julia and Borel DirectionsFilling Disks    §8.Closeness of a-Points    §9.Value Distribution of Derivatives of Meromorphic Functions    §10.ValueDistribution with Respect to Arguments  §11.ValueDistribution of Special Classes of Meromorphic Functions  §12.Entire CurvesChapter 6.Entire and Meromorphic Solutions of Ordinary Differential Equations(A.E.Eremenko)    §1.NonHnear ADEs with Meromorphic Solutions  §2.Linear Differential EquationsChapter 7.Some Applications of the Theory of Entire Fauctilnm(i.V.Ostrovskii)  §1.Riemann’B Boundary Problem with Infinite Index  §2.The Arithmetic of Probability Distributions  §3.Entire Characteristic and Ridge FunctionsReferences

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