复函数论导论

Preface

Ⅰ The Complex Number System

1 The Algebra and Geometry of Complex Numbers

1.1 The Field of Complex Numbers

1.2 Conjugate， Modulus， and Argument

2 Exponentials and Logarithms of Complex Numbers

2.1 Raising e to Complex Powers

2.2 Logarithms of Complex Numbers

2.3 Raising Complex Numbers to Complex Powers

3 Functions of a Complex Variable

3.1 Complex Functions

3.2 Combining Functions

3.3 Functions as Mappings

4 Exercises for Chapter Ⅰ

Ⅱ The Rudiments of Plane Topology

1 Basic Notation and Terminology

1.1 Disks

1.2 Interior Points， Open Sets

1.3 Closed Sets

1.4 Boundary， Closure，Interior

1.5 Sequences

1.6 Convergence of Complex Sequences

1.7 Accumulation Points of Complex Sequences

2 Continuity and Limits of Functions

2.1 Continuity

2.2 Limits of Functions

3 Connected Sets

3.1 Disconnected Sets

3.2 Connected Sets

3.3 Domains

3.4 Components of Open Sets

4 Compact Sets

4.1 Bounded Sets and Sequences

4.2 Cauchy Sequences

4.3 Compact Sets

4.4 Uniform Continuity

5 Exercises for Chapter Ⅱ

Ⅲ Analytic Functions

1 Complex Derivatives

1.1 Differentiability

1.2 Differentiation Rules

1.3 Analytic Functions

2 The Cauchy-Riemann Equations

2.1 The Cauchy-Riemann System of Equations

2.2 Consequences of the Cauchy-Riemann Relations

3 Exponential and Trigonometric Functions

3.1 Entire Functions

3.2 Trigonometric Functions

3.3 The Principal Arcsine and Arctangent Functions

4 Branches oflnverse Functions

4.1 Branches of lnverse Functions

4.2 Branches of the pth-root Function

4.3 Branches of the Logarithm Function

4.4 Branches of the A-power Function

5 Differentiability in the Real Sense

5.1 Real Differentiability

5.2 The Functions fx and fz

6 Exercises for Chapter Ⅲ

Ⅳ Complex lntegration

1 Paths in the Complex Plane

1.1 Paths

1.2 Smooth and Piecewise Smooth Paths

1.3 Parametrizing Line Segments

1.4 Reverse Paths， Path Sums

……

Ⅴ Cauchy's Theorem and its Consequences

Ⅵ Harmonic Functions

Ⅶ Sequences and Series of Analytic Functions

Ⅷ Isolated Singularities of Analytic Functions

Ⅸ Conformal Mapping

Ⅹ Constructing Analytic functions

Appendix A Background on Fields

Appendix B Winding Numbers Revisited

Index

The book at hand has its origins in and reflects the structure of a course that I have given regularly over the years at the University of Texas. The course in question is an undergraduate honors course in complex analysis. Its subscribers are for the most part math and physics majors， but a smattering of engineering students， those interested in a more substantial and more theoretically oriented introduction to the subject than our normal undergraduate complex variables course offers， can usually be found in the class. My approach to the course has been from its inception to teach it in everything save scope like a beginning graduate course in complex function theory. （To be honest， I have included some material in the book that I do not ordinarily cover in the course， this with the admitted purpose of making the book a suitable text for a first course in complex analysis at the graduate level.） Thus， the tone of the course is quite rigorous， while its pace is rather deliberate. Faced with a clientele that is bright， but mathematically less sophisticated than， say， a class of mathematics graduate students would be， I considered it imperative to give students access to a complete written record of the goings-on in my lectures， one containing full details of proofs that I might only sketch in class， the accent there being on the central idea involved in an argument rather than on the nitty-gritty technicalities of the proof. I also deemed it wise to provide the students with a generous supply of worked-out examples appropriate to the lecture material. Since none of the textbooks available when I started teaching the course had exactly the emphasis I was looking for， l began to compile my own set of lecture notes. It is these notes that have evolved into the present
　　book.