复分析综合大纲

浏览次数：332

《复分析》教学大纲

一、课程基本信息

开课单位:	数学科学研究所	课程代码:	MATH1221
课程名称:	复分析	英文名称:	Complex Analysis
学分:	3	学时:	48
授课对象:		授课语言:	中英文
先修课程:	数学分析II

二、课程简介和教学目的

This is a course on complex analysis at the advanced undergraduate level. We plan to include the following important properties of holomorphic functions: the power series expansion, Cauchy’s theorem, the conformal map, the calculus of residues, and the uniqueness of analytic continuation. If time permits, we will also talk about various topics about analytic functions. It is assumed that the students, who will take this class, have been trained in mathematical analysis and linear algebra.

三、教学内容、教学方式和学时安排

(each table is meant for 1.5 week, 6 periods )

Chapter 1	Complex Functions
1.1	Complex Valued Functions
1.2	Complex Differentiability
1.3	The Cauchy-Rieman Equations
1.4	Angles Under Holomorphic Maps

Chapter 2	Power Series
2.1	Formal Power Series
2.2	Convergent Power Series
2.3	Relations Between Formal and Convergent Power Series
2.4	Differentiation of Power Series

Chapter 3	Power Series Continued
3.1	Analytic Functions
3.2	The Inverse and Open Mapping Theorems
3.3	The Local Maximum Modulus Principle

Chapter 4	Cauchy’s Theorem
4.1	Holomorphic Functions
4.2	Integrals Over Paths
4.3	Local Primitive for a Holomorphic Function
4.4	Another Description of the Integral Along a Path

Chapter 5	Cauchy’s Theorem Continued
5.1	The Homotopy Form of Cauchy’s Theorem
5.2	Existence of Global Primitives
5.3	The Local Cauchy Formula

Chapter 6	Winding Numbers
6.1	The Winding Numbers
6.2	The Global Cauchy Theorem
6.3	Artin’s Proof

Chapter 7	Application of Cauchy’s Integral Formula
7.1	Uniform Limits of Analytic Functions
7.2	Laurent Series
7.3	Isolated Singularities

Chapter 8	Calculus of Residues
8.1	The Residue Formula
8.2	Evaluations of Definite Integrals
8.3	Fourier Transforms
8.4	Mellin Transforms

Chapter 9	Conformal Mappings
9.1	Schwarz Lemma
9.2	Analytic Automorphisms of the Disk
9.3	The Upper Half Plane

Chapter 10	Conformal Mappings Continued
10.1	Other Examples
10.2	Fractional Linear Transformations

Chapter 11	Harmonic Functions
11.1	Definitions
11.2	Examples
11.3	Basic Properties of Harmonic Functions
11.4

Chapter 12	Harmonic Functions Continued
12.1	The Poisson Formula
12.2	Construction of Harmonic Functions
12.3	( Differentiating under the Integral sign )

Chapter 13	Various Analytic Topics
13.1	Jensen’s Formula
13.2	The Picard-Borel Theorem

Chapter 14	Various Analytic Topics Continued
14.1	Bounds by the Real Part
14.2	The Use of Three Circles

Final Exam	The 17th. Week

四、推荐教材

书名	作者	译者	出版社	出版时间	ISBN
Complex Analysis	Eberhard Freitag, Rolf Busam		Springer; 2nd Edition (2009)		978-3-540-93982-5

五、参考书目

书名	作者	译者	出版社	出版时间	ISBN
Complex Analysis	Serge Lang		Springer; 4th Edition		0-387-98592-1