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Complex Analysis

Joseph Bak and Donald J. Newman
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
Allen Stenger
, on
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 ez, 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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