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

前言/序言

　　This book provides an introduction to complex analysis for students with　some familiarity with complex numbers from high school. Students should be familiar with the Cartesian representation of complex numbers and with the algebra of complex numbers， that is， they should know that i2=-1. A familiarity with multivariable calculus is also required， but here the fundamental ideas are reviewed. In fact， complex analysis provides a good training ground for multivariable calculus.It allows students to consolidate their understanding of parametrized curves， tangent vectors， arc length，gradients， line integrals， independence of path， and Greens theorem. The ideas surrounding independence of path are particularly difficult for students in calculus， and they are not absorbed by most students until they are seen again in other courses.
　　The book consists of sixteen chapters， which are divided into three parts.The first part， Chapters I-VII， includes basic material covered in all undergraduate courses. With the exception of a few sections， this material is much the same as that covered in Cauchys lectures， except that the emphasis on viewing functions as mappings reflects Riemanns influence. The second part， Chapters VIII-XI， bridges the nineteenth and the twentieth centuries. About half this material would be covered in a typical undergraduate course， depending upon the taste and pace of the instructor. The material on the Poisson integral is of interest to electrical engineers， while the material on hyperbolic geometry is of interest to pure mathematicians and also to high school mathematics teachers. The third part， Chapters XII-XVI， consists of a careful selection of special topics that illustrate the scope and power of complex analysis methods. These topics include Julia sets and the Mandelbrot set， Dirichlet series and the prime number theorem， and the uniformization theorem for Riemann surfaces. The final five chapters serve also to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis.

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