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研究生课程《单叶函数理论》教学大纲

  发布日期:2016-12-16  浏览量:172


课程编号:Math2079

课程名称:单叶函数理论

英文名称:Univalent functions

                                

开课单位:澳门赌搏网站大全                         

开课学期:秋

课内学时:36                                   

教学方式:讲授

适用专业及层次:数学专业硕士研究生             

考核方式:考试

预修课程:复变函数

 

一、教学目标与要求

本课程是澳门赌搏网站大全复分析方向上研究生的必修核心专业课程。其教学目的是向学生先容单叶函数的基本理论。通过本课程的学习,要求研究生掌握单叶函数理论的基本理论和方法,为以后的继续学习和开展科学研究打好基础。

 

二、课程内容与学时分配

    Chapter 1  Geometric function theory9 hours

        11  basic principles                      

        12  local mapping properties

        13  normal families                       

        14  extremal problems

        15  the  Riemann mapping theorem        

        16  analytic continuation

        1.  7  harmonic and subharmonic functions    

        1.  8  Green’s functions

        1.  9  positive harmonic functions

 

Chapter 2  Elementary theory of univalent functions6 hours

        21  introduction                                                  22  the area theorem

        23  growth and distortion theorems                     24  coefficient estimates

        25  convex and starlike functions                         26  close-to-convex functions

        27  typically real functions

 

Chapter 3  parametric representation of slit mappings6 hours

        31  Caratheodory convergence theorem              32  density of slit mappings

        33  Loewner’s Differential equation                     34  Univalence of solutions

        35  the third coefficient                                       36  Radius of starlikeness

 

Chapter 4  Subordination 5 hours

        41  basic priciples                       42  coefficient inequalities

        43  sharpened forms of the Schwarz lemma                    44  majorization

        45  univalent subordinate functions       

   

Chapter 5   Some special topics5 hours

        51  bounded univalent functions  

        52 sections of univalent functions

        53  convolutions of convex functions                

        54  coefficient multipliers

        55  criteria for univalence

 

Chapter 6  General extremal problems5 hours

        61  functiions on linear spaces                

        62  representation of linear functionals

        63  extreme points and support points     

        64  properties of extremal functions

        65  extreme points of S

 

三、教材                  

Peter L. Duren,  Univalent functions, Springer-Verlag, New York, Berlin, Hiedelberg and Tokyo, 1983.

四,主要参考书

1A. W. Goodman,  Univalent Functions, Mariner Publishing Co. Inc, 1983

 

大纲撰写负责人: 龙波涌                     授课教师:龙波涌,黄华鹰等

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