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