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Function Theory of One Complex Variable

Robert E. Greene and Steven G. Krantz
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 40
[Reviewed by
Tom Schulte
, on

This book is Volume 40 in the American Mathematical Society series “Graduate Studies in Mathematics.” But let that not daunt anyone seeking an introduction to complex analysis. This text introduces this beautiful subject starting from the most elementary properties of the complex numbers. Familiarity with just the basic definition of a complex number as having a real and imaginary part is enough to get started. Of course, things ramp up rather quickly and by the end of the book the reader will be considering the Bieberbach Conjecture and roaming in Hilbert Space. 

This would make a good textbook for a first-year graduate course in complex analysis that starts around chapter three, a text that handily contains review material for anyone needing to recall more elementary points. No solutions are provided for the problems, so this may not be a good volume for self-guided exploration. I am also surprised that the authors did not use a page or two to summarize and collect in one place the notation introduced to the reader over the course of nearly five hundred pages.

Function Theory of One Complex Variable is written in a style that is lively and engaging without being breezy. Bringing in “physical intuition” during explanations and nodding to the fact that “making the questions themselves precise takes some thought” is the type of language approach that engages the reader in this book. There are 75 illustrative figures. Example and proofs are very frequent in the course of any given chapter. As a result, this book is a thorough explanation of the topics it covers, from cover to cover.

The authors illuminate points of complex analysis by comparing and contrasting complex analysis with its real variable counterpart. So, for instance, the subject of complex differentiability starts “recall from ordinary calculus… partial derivatives and directional derivates, as well as total derivatives. ” This helps make their textbook an excellent introduction from someone recently finished with their undergraduate calculus semesters.

After transitioning the reader from multivariable real calculus to complex analysis, the book concludes with several special topics chapters on special functions, the prime number theorem, and the Bergman kernel. The authors also cover Hp spaces, and the Bell-Ligocka approach to proving smoothness to the boundary of biholomorphic mappings.

Tom Schulte ( ) is a graduate student at Oakland University. He recently spent a Saturday with a truck and a two-hour (one-way) trip to accept a collection of old math books. Thus, several slide rule guides were saved from an ignominious end.

Preface to the Third Edition xiii

Preface to the Second Edition xv

Preface to the First Edition xvii

Acknowledgments xix

Chapter 1. Fundamental Concepts 1

1.1. Elementary Properties of the Complex Numbers 1

1.2. Further Properties of the Complex Numbers 3

1.3. Complex Polynomials 10

1.4. Holomorphic Functions, the Cauchy-Riemann Equations,

and Harmonic Functions 14

1.5. Real and Holomorphic Antiderivatives 17

Exercises 20

Chapter 2. Complex Line Integrals 29

2.1. Real and Complex Line Integrals 29

2.2. Complex Differentiability and Conformality 34

2.3. Antiderivatives Revisited 40

2.4. The Cauchy Integral Formula and the Cauchy

Integral Theorem 43


viii Contents

2.5. The Cauchy Integral Formula: Some Examples 50

2.6. An Introduction to the Cauchy Integral Theorem and the

Cauchy Integral Formula for More General Curves 53

Exercises 60

Chapter 3. Applications of the Cauchy Integral 69

3.1. Differentiability Properties of Holomorphic Functions 69

3.2. Complex Power Series 74

3.3. The Power Series Expansion for a Holomorphic Function 81

3.4. The Cauchy Estimates and Liouville’s Theorem 84

3.5. Uniform Limits of Holomorphic Functions 88

3.6. The Zeros of a Holomorphic Function 90

Exercises 94

Chapter 4. Meromorphic Functions and Residues 105

4.1. The Behavior of a Holomorphic Function Near an

Isolated Singularity 105

4.2. Expansion around Singular Points 109

4.3. Existence of Laurent Expansions 113

4.4. Examples of Laurent Expansions 119

4.5. The Calculus of Residues 122

4.6. Applications of the Calculus of Residues to the

Calculation of Definite Integrals and Sums 128

4.7. Meromorphic Functions and Singularities at Infinity 137

Exercises 145

Chapter 5. The Zeros of a Holomorphic Function 157

5.1. Counting Zeros and Poles 157

5.2. The Local Geometry of Holomorphic Functions 162

5.3. Further Results on the Zeros of Holomorphic Functions 166

5.4. The Maximum Modulus Principle 169

5.5. The Schwarz Lemma 171

Exercises 174

Contents ix

Chapter 6. Holomorphic Functions as Geometric Mappings 179

6.1. Biholomorphic Mappings of the Complex Plane to Itself 180

6.2. Biholomorphic Mappings of the Unit Disc to Itself 182

6.3. Linear Fractional Transformations 184

6.4. The Riemann Mapping Theorem: Statement and

Idea of Proof 189

6.5. Normal Families 192

6.6. Holomorphically Simply Connected Domains 196

6.7. The Proof of the Analytic Form of the Riemann

Mapping Theorem 198

Exercises 202

Chapter 7. Harmonic Functions 207

7.1. Basic Properties of Harmonic Functions 208

7.2. The Maximum Principle and the Mean Value Property 210

7.3. The Poisson Integral Formula 212

7.4. Regularity of Harmonic Functions 218

7.5. The Schwarz Reflection Principle 220

7.6. Harnack’s Principle 224

7.7. The Dirichlet Problem and Subharmonic Functions 226

7.8. The Perr`on Method and the Solution of the

Dirichlet Problem 236

7.9. Conformal Mappings of Annuli 240

Exercises 243

Chapter 8. Infinite Series and Products 255

8.1. Basic Concepts Concerning Infinite Sums and Products 255

8.2. The Weierstrass Factorization Theorem 263

8.3. The Theorems of Weierstrass and Mittag-Leffler:

Interpolation Problems 266

Exercises 274

Chapter 9. Applications of Infinite Sums and Products 279

x Contents

9.1. Jensen’s Formula and an Introduction to

Blaschke Products 279

9.2. The Hadamard Gap Theorem 285

9.3. Entire Functions of Finite Order 288

Exercises 296

Chapter 10. Analytic Continuation 299

10.1. Definition of an Analytic Function Element 299

10.2. Analytic Continuation along a Curve 304

10.3. The Monodromy Theorem 307

10.4. The Idea of a Riemann Surface 310

10.5. The Elliptic Modular Function and Picard’s Theorem 314

10.6. Elliptic Functions 323

Exercises 330

Chapter 11. Topology 335

11.1. Multiply Connected Domains 335

11.2. The Cauchy Integral Formula for Multiply

Connected Domains 338

11.3. Holomorphic Simple Connectivity and Topological

Simple Connectivity 343

11.4. Simple Connectivity and Connectedness of

the Complement 344

11.5. Multiply Connected Domains Revisited 349

Exercises 352

Chapter 12. Rational Approximation Theory 363

12.1. Runge’s Theorem 363

12.2. Mergelyan’s Theorem 369

12.3. Some Remarks about Analytic Capacity 378

Exercises 381

Chapter 13. Special Classes of Holomorphic Functions 385

13.1. Schlicht Functions and the Bieberbach Conjecture 386

Contents xi

13.2. Continuity to the Boundary of Conformal Mappings 392

13.3. Hardy Spaces 401

13.4. Boundary Behavior of Functions in Hardy Classes

[An Optional Section for Those Who Know

Elementary Measure Theory] 406

Exercises 412

Chapter 14. Hilbert Spaces of Holomorphic Functions, the Bergman

Kernel, and Biholomorphic Mappings 415

14.1. The Geometry of Hilbert Space 415

14.2. Orthonormal Systems in Hilbert Space 426

14.3. The Bergman Kernel 431

14.4. Bell’s Condition R 438

14.5. Smoothness to the Boundary of Conformal Mappings 443

Exercises 446

Chapter 15. Special Functions 449

15.1. The Gamma and Beta Functions 449

15.2. The Riemann Zeta Function 457

Exercises 467

Chapter 16. The Prime Number Theorem 471

16.0. Introduction 471

16.1. Complex Analysis and the Prime Number Theorem 473

16.2. Precise Connections to Complex Analysis 478

16.3. Proof of the Integral Theorem 483

Exercises 485

APPENDIX A: Real Analysis 487

APPENDIX B: The Statement and Proof of Goursat’s Theorem 493

References 497

Index 501