Course Description

 

Complex variable function was born in eighteenth Century, mainly established by Euler, D'Alembert, Laplace and other mathematicians. By nineteenth Century, due to the work of Cauchy, Weierstrass, Riemann and other mathematicians, the theory of complex variable function has been fully developed, and become a very popular new branch of mathematics. At the beginning of twentieth Century, Swedish mathematician Leffler, French mathematician Poincare, Hadamard have done a lot of research work, to develop the complex variable function theory of a broader research area. Today the complex function has a history of more than and 300 years, widely used in many natural science fields, such as theoretical physics, aerodynamics, fluid mechanics, elastic mechanics, automatic control theory, signal processing, electronic engineering, quantum information and quantum computing. With its perfect theoretical system and unique techniques, Complex function theory which promotes the development of many disciplines has become an important part of mathematics of complex function, and a powerful tool to solve some theoretical and practical problems.

 

 

 

 

I   Property and teaching aim of the course

Function Theory of one Complex Variable based on Mathematical Analysis, is a branch of analytical mathematics and devoted to the study of complex-valued analytic functions of one complex variable in teams of the methods of limits and integrals. This course is very important for the study of Functional Analysis，Differential Equations and so on.

   The purpose of this course is to introduce the basic concepts, methods, theories and related applications of complex analysis so that the students can understand well the basic contents of the course and establish a stability foundation for the studying of the subsequence courses.

 

II           Teaching contents

(I)                     Complex Number Fields

1．Complex numbers and their operations;
2．Complex plane, moduli and arguments of complex numbers;
3．Roots of complex numbers and applications;
4．Point sets and regions on the complex plane.

(II)                   Analytic Functions

1．Definitions of complex variable functions and their mapping properties;
2.Limits , continuity and derivatives of complex variable functions;
3．Cauchy-Riemann equations(C-R conditions);
4．Concepts and basic properties of analytic functions.

(III)                Elementary functions

1．Definitions of Exponential functions and their basic properties;
2．Concepts and basic properties and related identities of logarithmic functions;
3．Complex power functions and related properties;
4．Concepts and related identities of trigonometric functions.

(IV)                Integrals

1．Definitions and basic properties of integrals of complex variable functions;
2．Cauchy Integral Theorem;
3．Cauchy Integral Formula;
4．Derivatives of high order formula.

(V)                   Series

1．Definitions and convergence of series of complex numbers;
2．Taylor expansion;
3．Laurent expansion;
4．Convergence disk, convergence radius of power series.

(VI)                Residues and Poles

1．Definitions and calculations of residues;
2．The three types of isolated singular points;
3．Residues at poles and applications;
4．Zeros of analytic functions.

(VII)              Applications of Reside

1．Evaluation of improper integrals;
2．Arguments Principle, Rouche's Theorem.

 

 

 

 

 

复变函数产生于十八世纪，主要由欧拉、达朗贝尔、拉普拉斯等数学家创建。到十九世纪，由于柯西、维尔斯特拉斯、黎曼等数学家的工作，使得复变函数理论得到全面发展，并变成十分热门的新数学分支。二十世纪初，瑞典数学家列夫勒、法国数学家彭加勒、阿达玛等作了大量的研究工作，开拓了复变函数理论更广阔的研究领域。到今天复变函数已有三百多年的历史，被广泛应用于自然科学的众多领域，如理论物理、空气动力学、流体力学、弹性力学、自动控制学、信号处理、电子工程、量子信息与量子计算等领域。复变函数论以其完美的理论体系与独特的技巧方法成为数学学科的一个重要组成部分，推动了许多学科的发展，成为解决某些理论与实际问题的强有力工具。

 

一、本课程的性质和任务

《复变函数论》是在《数学分析》的基础上,应用分析与积分方法研究单复变量复值解析函数的一门分析数学,它是学习与研究《泛函分析》、《微分方程》等课程的重要基础.

本课程的教学目的及任务是向学生介绍复变函数论的基本概念、基本方法、基本理论以及相关应用,使学生对复变函数理论的基本内容有一个初步的了解,为进一步学习后继课程打下坚实的基础.

 

二、本课程的基本内容

（一）复数域

1、复数及其运算；

2、复平面、复数的模与辐角；

3、复数的求根公式及应用；

4、复平面上的点集、区域。

（二）解析函数

1、复变函数的定义和其映射性质；

2、复变函数的极限、连续、导数；

3、柯西－黎曼条件（C-R条件）；

4、解析函数的概念及基本性质。

（三）初等函数

1、指数函数的定义及基本性质；

2、对数函数的定义、性质及其相关恒等式；

3、复幂函数和其相关性质；

4、三角函数定义、相关等式。

（四）积分

1、复变函数积分的定义及基本性质；

2、柯西积分定理；

3、柯西积分公式；

4、高阶导数公式。

（五）级数

1、复数项级数的定义、收敛性；

2、泰勒展式；

3、罗朗展式；

4、幂级数的收敛域、收敛半径、和函数的解析性。

（六）留数

1、留数的定义和计算；

2、孤立奇点及其分类；

3、极点处的留数及相关应用；

4、解析函数的零点。

（七）留数的应用

1、瑕积分的计算；

2、辐角原理、儒歇定理。