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

*ｧ*1.1. Elementary Properties of the Complex Numbers 1

*ｧ*1.2. Further Properties of the Complex Numbers 3

*ｧ*1.3. Complex Polynomials 10

*ｧ*1.4. Holomorphic Functions, the Cauchy-Riemann Equations,

and Harmonic Functions 14

*ｧ*1.5. Real and Holomorphic Antiderivatives 17

Exercises 20

Chapter 2. Complex Line Integrals 29

*ｧ*2.1. Real and Complex Line Integrals 29

*ｧ*2.2. Complex Differentiability and Conformality 34

*ｧ*2.3. Antiderivatives Revisited 40

*ｧ*2.4. The Cauchy Integral Formula and the Cauchy

Integral Theorem 43

vii

viii *Contents*

*ｧ*2.5. The Cauchy Integral Formula: Some Examples 50

*ｧ*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

*ｧ*3.1. Differentiability Properties of Holomorphic Functions 69

*ｧ*3.2. Complex Power Series 74

*ｧ*3.3. The Power Series Expansion for a Holomorphic Function 81

*ｧ*3.4. The Cauchy Estimates and Liouville’s Theorem 84

*ｧ*3.5. Uniform Limits of Holomorphic Functions 88

*ｧ*3.6. The Zeros of a Holomorphic Function 90

Exercises 94

Chapter 4. Meromorphic Functions and Residues 105

*ｧ*4.1. The Behavior of a Holomorphic Function Near an

Isolated Singularity 105

*ｧ*4.2. Expansion around Singular Points 109

*ｧ*4.3. Existence of Laurent Expansions 113

*ｧ*4.4. Examples of Laurent Expansions 119

*ｧ*4.5. The Calculus of Residues 122

*ｧ*4.6. Applications of the Calculus of Residues to the

Calculation of Definite Integrals and Sums 128

*ｧ*4.7. Meromorphic Functions and Singularities at Infinity 137

Exercises 145

Chapter 5. The Zeros of a Holomorphic Function 157

*ｧ*5.1. Counting Zeros and Poles 157

*ｧ*5.2. The Local Geometry of Holomorphic Functions 162

*ｧ*5.3. Further Results on the Zeros of Holomorphic Functions 166

*ｧ*5.4. The Maximum Modulus Principle 169

*ｧ*5.5. The Schwarz Lemma 171

Exercises 174

*Contents* ix

Chapter 6. Holomorphic Functions as Geometric Mappings 179

*ｧ*6.1. Biholomorphic Mappings of the Complex Plane to Itself 180

*ｧ*6.2. Biholomorphic Mappings of the Unit Disc to Itself 182

*ｧ*6.3. Linear Fractional Transformations 184

*ｧ*6.4. The Riemann Mapping Theorem: Statement and

Idea of Proof 189

*ｧ*6.5. Normal Families 192

*ｧ*6.6. Holomorphically Simply Connected Domains 196

*ｧ*6.7. The Proof of the Analytic Form of the Riemann

Mapping Theorem 198

Exercises 202

Chapter 7. Harmonic Functions 207

*ｧ*7.1. Basic Properties of Harmonic Functions 208

*ｧ*7.2. The Maximum Principle and the Mean Value Property 210

*ｧ*7.3. The Poisson Integral Formula 212

*ｧ*7.4. Regularity of Harmonic Functions 218

*ｧ*7.5. The Schwarz Reflection Principle 220

*ｧ*7.6. Harnack’s Principle 224

*ｧ*7.7. The Dirichlet Problem and Subharmonic Functions 226

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

Dirichlet Problem 236

*ｧ*7.9. Conformal Mappings of Annuli 240

Exercises 243

Chapter 8. Infinite Series and Products 255

*ｧ*8.1. Basic Concepts Concerning Infinite Sums and Products 255

*ｧ*8.2. The Weierstrass Factorization Theorem 263

*ｧ*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*

*ｧ*9.1. Jensen’s Formula and an Introduction to

Blaschke Products 279

*ｧ*9.2. The Hadamard Gap Theorem 285

*ｧ*9.3. Entire Functions of Finite Order 288

Exercises 296

Chapter 10. Analytic Continuation 299

*ｧ*10.1. Definition of an Analytic Function Element 299

*ｧ*10.2. Analytic Continuation along a Curve 304

*ｧ*10.3. The Monodromy Theorem 307

*ｧ*10.4. The Idea of a Riemann Surface 310

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

*ｧ*10.6. Elliptic Functions 323

Exercises 330

Chapter 11. Topology 335

*ｧ*11.1. Multiply Connected Domains 335

*ｧ*11.2. The Cauchy Integral Formula for Multiply

Connected Domains 338

*ｧ*11.3. Holomorphic Simple Connectivity and Topological

Simple Connectivity 343

*ｧ*11.4. Simple Connectivity and Connectedness of

the Complement 344

*ｧ*11.5. Multiply Connected Domains Revisited 349

Exercises 352

Chapter 12. Rational Approximation Theory 363

*ｧ*12.1. Runge’s Theorem 363

*ｧ*12.2. Mergelyan’s Theorem 369

*ｧ*12.3. Some Remarks about Analytic Capacity 378

Exercises 381

Chapter 13. Special Classes of Holomorphic Functions 385

*ｧ*13.1. Schlicht Functions and the Bieberbach Conjecture 386

*Contents* xi

*ｧ*13.2. Continuity to the Boundary of Conformal Mappings 392

*ｧ*13.3. Hardy Spaces 401

*ｧ*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

*ｧ*14.1. The Geometry of Hilbert Space 415

*ｧ*14.2. Orthonormal Systems in Hilbert Space 426

*ｧ*14.3. The Bergman Kernel 431

*ｧ*14.4. Bell’s Condition *R* 438

*ｧ*14.5. Smoothness to the Boundary of Conformal Mappings 443

Exercises 446

Chapter 15. Special Functions 449

*ｧ*15.1. The Gamma and Beta Functions 449

*ｧ*15.2. The Riemann Zeta Function 457

Exercises 467

Chapter 16. The Prime Number Theorem 471

*ｧ*16.0. Introduction 471

*ｧ*16.1. Complex Analysis and the Prime Number Theorem 473

*ｧ*16.2. Precise Connections to Complex Analysis 478

*ｧ*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