复分析
豆瓣
加默兰
简介
《复分析》从现代数学的观点介绍复分析的基本知识与常用工具，全书共分为8章，主要包括：复数、复函数、作为映射的解析函数、复积分、级数与乘积展开、共形映射，软件克雷问题、椭圆函数以及全局解析函数，此外，大部分章节后都有练习，便于学生掌握书中内容。
contents
Preface
Introduction
FIRST PART
Chapter 1 The Complex Plane and Elementary Functions
1.Complex Numbers
2.Polar Representation
3.Stereographic Projection
4.The Square and Square Root Functions
5.The Exponential Function
6.The Logarithm Function
7.Power Functions and Phase Factors
8.Trigonometric and Hyperbolic Functions
Chapter 2 Analytic Functions
1.Review of Basic Analysis
2.Analytic Functions
3.The CauChy-Riemann Equations
4.Inverse Mappings and the Jacobian
5.Harmonic Functions
6.Conformal Mappings
7.Fractional Linear Transformations
Chapter 3 Line Integrals and Harmonic Functions
1.Line Integrals and Greens Theorem
2.Independence of Path
3.Harmonic Conjugates
4.The Mean Value Property
5.The Maximum Principle
6.Applications to Fluid Dynamics
7.Other Applications to Physics
Chapter 4 Complex Integration and Analyticity
1.Complex Line Integrals
2.Fundamental Theorem of Calculus for Analytic Functions
3.Cauchys Theorem
4.The Cauchy Integral Formula
5.Liouvilles Theorem
6.Moreras Theorem
7.Goursats Theorem
8.Complex Notation and Pompeius Formula
Chapter 5 Power Series
1.Infinite Series
2.Sequences and Series of Functions
3.Power Series
4.Power Series Expansion of an Analytic Function
5.Power Series Expansion at Infinity
6.Manipulation of Power Series
7.The Zeros of an Analytic Function
8.Analytic Continuation
Chapter 6 Laurent Series and Isolated Singularities
1.The Laurent Decomposition
2.Isolated Singularities of an Analytic Function
3.Isolated Singularity at Infinity
4.Partial Fractions Decomposition
5.Periodic Functions
6.Fourier Series
Chapter 7 The Residue Calculus
1.The Residue Theorem
2.Integrals Featuring Rational Functions
3.Integrals of Trigonometric Functions
4.Integrands with Branch Points
5.Fractional Residues
6.Principal Values
7.Jordans Lemma
8.Exterior Domains
SECOND PART
Chapter 8 The Logarithmic Integral
1.The Argument Principle
2.Rouches Theorem
3.Hurwitzs Theorem
4.Open Mapping and Inverse Function Theorems
5.Critical Points
6.Winding Numbers
……
THIRD PART
Hints and Solutions for Selected Exercises
References
List of Symbols
Index
