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Publisher:

Springer

Publication Date:

1999

Number of Pages:

304

Format:

Hardcover

Edition:

2

Series:

Undergraduate Texts in Mathematics

Price:

79.95

ISBN:

9780387947563

Category:

Textbook

[Reviewed by , on ]

Allen Stenger

10/15/2009

This is a clever, concise, concrete, and classical complex analysis book, aimed at undergraduates with no background beyond single-variable calculus. The book has an eclectic flavor; rather than develop any general theories, the authors work toward a number of classical results, and usually take the shortest path to get there.

The book generally takes an analytic rather than a geometric approach; the Cauchy-Riemann equations are central. It works up to analytic functions by going through polynomials and entire functions, and only then considers functions analytic on a disk and then analytic on a region. The elementary functions are developed as extensions of those functions on the reals, rather than as power series. Similarly, the book starts with polygonal paths with only horizontal or vertical segments, and works up to general curves. There are no Riemann surfaces, and multi-valued functions such as the logarithm are sidestepped by explicitly defining a useful branch and showing that it has the desired properties.

The book has a modest number of applications, including some discussion of fluid flow and the Riemann mapping theorem. Most of the applications are to other branches of mathematics rather than to other sciences, and cover fields such as combinatorics and evaluation of definite integrals and infinite series. There is even has a complete proof of the prime number theorem.

There are many exercises, and they cover a wide range of difficulty, from routine applications of techniques in the text through quite challenging problems. Answers to all exercises are given in the back of the book, although usually they are sketches of the answer in a couple of sentences rather than a detailed answer.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

Preface

1. The Complex Numbers

Introduction

1.1. The Field of Complex Numbers

1.2. The Complex Plane

1.3. Topological Aspects of the Complex Plane

1.4. Stereographic Projection; The Point at Infinity

Exercises

2. Functions of the Complex Variable z

Introduction

2.1. Analytic Polynomials

2.2. Power Series

2.3. Differentiability and Uniqueness of Power Series

Exercises

3. Analytic Functions

3.1. Analyticity and the Cauchy-Riemann Equations

3.2. The Functions e^{z}, sin z, cos z

Exercises

4. Line Integrals and Entire Functions

Introduction

4.1. Properties of the Line Integral

4.2. The Closed Curve Theorem for Entire Functions

Exercises

5. Properties of Entire Functions

5.1. The Cauchy Integral Formula and Taylor Expansion for Entire Functions

5.2. Liouville Theorems and the Fundamental Theorem of Algebra

Exercises

6. Properties of Analytic Functions

Introduction

6.1. The Power Series Representation for Functions Analytic in a Disc

6.2. Analyticity in an Arbitrary Open Set

6.3. The Uniqueness, Mean-Value, and Maximum-Modulus Theorems

Exercises

7. Further Properties of Analytic Functions

7.1. The Open Mapping Theorem; Schwarz’ Lemma

7.2. The Converse of Cauchy’s Theorem: Morera’s Theorem; The Schwarz Reflection Principle

Exercises

8. Simply Connected Domains

8.1. The General Cauchy Closed Curve Theorem

8.2. The Analytic Function Log z

Exercises

9. Isolated Singularities of an Analytic Function

9.1. Classification of Isolated Singularities; Riemann’s Principle and the Casorati-Weierstrass Theorem

9.2. Laurent Expansions

Exercises

10. The Residue Theorem

lO.1. Winding Numbers and the Cauchy Residue Theorem

lO.2. Applications of the Residue Theorem

Exercises

11. Applications of The Residue Theorem to the Evaluation of Integrals and Sums

Introduction

11.1. Evaluation of Definite Integrals by Contour Integral Techniques

11.2. Application of Contour Integral Methods to Evaluation and Estimation of Sums

Exercises

12. Further Contour Integral Techniques

12.1. Shifting the Contour of Integration

12.2. An Entire Function Bounded in Every Direction

Exercises

13. Introduction to Conformal Mapping

13.1. Conformal Equivalence

13.2. Special Mappings

Exercises

14. The Riemann Mapping Theorem

14.1. Conformal Mapping and Hydrodynamics

14.2. The Riemann Mapping Theorem

Exercises

15. Maximum-Modulus Theorems for Unbounded Domains

15.1. A General Maximum-Modulus Theorem

15.2. The Phragmén-Lindelöf Theorem

Exercises

16. Harmonic Functions

16.1. Poisson Formulae and the Dirichlet Problem

16.2. Liouville Theorems for **Re** f; Zeroes of Entire Functions of Finite Order

Exercises

17. Different Forms of Analytic Functions

Introduction

17.1. Infinite Products

17.2. Analytic Functions Defined by Definite Integrals

Exercises

18. Analytic Continuation; The Gamma and Zeta Functions

Introduction

18.1. Power Series

18.2. The Gamma and Zeta Functions

Exercises

19. Applications to Other Areas of Mathematics

Introduction

19.1. A Partition Problem

19.2. An Infinite System of Equations

19.3. A Variation Problem

19.4. The Fourier Uniqueness Theorem

19.5. The Prime-Number Theorem

Exercises

Appendices

1. A Note on Simply Connected Regions

II. Circulation and Flux as Contour Integrals

III. Steady-State Temperatures; The Heat Equation

IV. Tchebychev Estimates

Answers

Bibliography

Index

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