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研究生课程《调和映射理论》教学大纲

  发布日期:2016-12-16  浏览量:190


课程编号:Math2080

课程名称:调和映射理论

英文名称:harmonic mappings in the plane

                                

课单位:澳门赌搏网站大全                          

开课学期:春

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,单叶函数理论

 

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平面上调和映射理论的基本理论。通过本课程的学习,要求研究生掌握平面上调和映射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  Preliminaries5 hours

        11  Harmonic Mappings                       12  some basic facts

        13  the argument principle                     14  the dirichlet problem

        15  conformal Mappings              16  overview of harmonic mappings theory

 

Chapter 2  Ceneral properties of harmonic mappings6 hours

        21  cirtical points of harmonic functions         

        22  lewy’s Theorem

        23  Heinz’s Theorem                         

        24  Rado’s Theorem

        25  Counterexamples in higher dimensions            

        26  Approximation theorem

     

 Chapter 3  harmonc mappings onto convex regions6 hours

        31  the rado-Kneser-Choquet Theorem                   

        32  Choqet’s proof

        33  boundary behavior                            

        34  the shear construction

        35  structure of convex mappings

        36  Coveing Theorems and coefficient bounds

 

Chapter 4   Harmonic self-mappings of the disk6 hours

        41  representation by Rado-Kneser-Choquet Theorem      

        42  mappings onto regular polygons

        43  arbitrary convex polygons                 

        44  sharp form of Heinz’s inequality

        45  coefficient estimates       

        46  Schwarz’s Lemma for harmonic mappings

 

Chapter 5   Harmonic univalent functions6 hours

        51  normalizations  

        52  normal families

        53  the harmonic Koebe function                  

        54  coefficient conjectures

 

Chapter 6   Extremal problems7 hours

        61  minimum area                            

        62 covering theorems

        63  estimation of the second coeffecient            

        64  growth and distortion

        65  Marty relation                            

        66  typically real functions

        67  starlike functions 

 

三、教材                      

Peter Duren,   Harmoic mappings in the plane,  Cambridge university Press,  2004

 

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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