Foreword ........................................................ v
Prerequisites .................................................. ix
PART ONE
Basic Theory .................................................... 1
CHAPTER I Complex Numbers and Functions ......................... 3
§1. Definition .................................................. 3
§2. Polar Form .................................................. 8
§3. Complex Valued Functions ................................... 12
§4. Limits and Compact Sets .................................... 17
Compact Sets ............................................... 21
§5. Complex Differentiability .................................. 27
§6. The Cauchy-Riemann Equations ............................... 31
§7. Angles Under Holomorphic Maps .............................. 33
CHAPTER II Power Series ........................................ 37
§1. Formal Power Series ........................................ 37
§2. Convergent Power Series .................................... 47
§3. Relations Between Formal and Convergent Series ............. 60
Sums and Products ....................................... 60
Quotients ............................................... 64
Composition of Series ................................... 66
§4. Analytic Functions ......................................... 68
§5. Differentiation of Power Series ............................ 72
§6. The Inverse and Open Mapping Theorems ...................... 76
§7. The Local Maximum Modulus Principle ........................ 83
CHAPTER III Cauchy's Theorem, First Part ....................... 86
§1. Holomorphic Functions on Connected Sets .................... 86
Appendix: Connectedness ................................. 92
§2. Integrals Over Paths ....................................... 94
§3. Local Primitive for a Holomorphic Function ................ 104
§4. Another Description of the Integral Along a Path .......... 110
§5. The Homotopy Form of Cauchy's Theorem ..................... 115
§6. Existence of Global Primitives. Definition of the
Logarithm ................................................. 119
§7. The Local Cauchy Formula .................................. 125
CHAPTER IV Winding Numbers and Cauchy's Theorem ............... 133
§1. The Winding Number ........................................ 134
§2. The Global Cauchy Theorem ................................. 138
Dixon's Proof of Theorem 2.5 (Cauchy's Formula) ........ 147
§3. Artin's Proof ............................................. 149
CHAPTER V Applications of Cauchy's Integral Formula ........... 156
§1. Uniform Limits of Analytic Functions ...................... 156
§2. Laurent Series ............................................ 161
§3. Isolated Singularities .................................... 165
Removable Singularities ................................ 165
Poles .................................................. 166
Essential Singularities ................................ 168
CHAPTER VI Calculus of Residues ............................... 173
§1. The Residue Formula ....................................... 173
Residues of Differentials .............................. 184
§2. Evaluation of Definite Integrals .......................... 191
Fourier Transforms ..................................... 194
Trigonometric Integrals ................................ 197
Mellin Transforms ...................................... 199
CHAPTER VII Conformal Mappings ................................ 208
§1. Schwarz Lemma ............................................. 210
§2. Analytic Automorphisms of the Disc ........................ 212
§3. The Upper Half Plane ...................................... 215
§4. Other Examples ............................................ 220
§5. Fractional Linear Transformations ......................... 231
CHAPTER VIII Harmonic Functions ............................... 241
§1. Definition ................................................ 241
Application: Perpendicularity .......................... 246
Application: Flow Lines ................................ 248
§2. Examples .................................................. 252
§3. Basic Properties of Harmonic Functions .................... 259
§4. The Poisson Formula ....................................... 271
The Poisson Integral as a Convolution .................. 273
§5. Construction of Harmonic Functions ........................ 276
§6. Appendix. Differentiating Under the Integral Sign ......... 286
PART TWO Geometric Function Theory ............................ 291
CHAPTER IX Schwarz Reflection ................................. 293
§1. Schwarz Reflection (by Complex Conjugation) ............... 293
§2. Reflection Across Analytic Arcs ........................... 297
§3. Application of Schwarz Reflection ......................... 303
CHAPTER X The Riemann Mapping Theorem ......................... 306
§1. Statement of the Theorem .................................. 306
§2. Compact Sets in Function Spaces ........................... 308
§3. Proof of the Riemann Mapping Theorem ...................... 311
§4. Behavior at the Boundary .................................. 314
CHAPTER XI Analytic Continuation Along Curves ................. 322
§1. Continuation Along a Curve ................................ 322
§2. The Dilogarithm ........................................... 331
§3. Application to Picard's Theorem ........................... 335
PART THREE Various Analytic Topics ............................ 337
CHAPTER XII Applications of the Maximum Modulus Principle
and Jensen's Formula .............................. 339
§1. Jensen's Formula .......................................... 340
§2. The Picard-Borel Theorem .................................. 346
§3. Bounds by the Real Part, Borel-Caratheodory Theorem ....... 354
§4. The Use of Three Circles and the Effect of Small
Derivatives ............................................... 356
Hermite Interpolation Formula .......................... 358
§5. Entire Functions with Rational Values ..................... 360
§6. The Phragmen-Lindelof and Hadamard Theorems ............... 365
CHAPTER XIII Entire and Meromorphic Functions ................. 372
§1. Infinite Products ......................................... 372
§2. Weierstrass Products ...................................... 376
§3. Functions of Finite Order ................................. 382
§4. Meromorphic Functions, Mittag-Leffler Theorem ............. 387
CHAPTER XIV Elliptic Functions ................................ 391
§1. The Liouville Theorems .................................... 391
§2. The Weierstrass Function .................................. 395
§3. The Addition Theorem ...................................... 400
§4. The Sigma and Zeta Functions .............................. 403
CHAPTER XV The Gamma and Zeta Functions ....................... 408
§1. The Differentiation Lemma ................................. 409
§2. The Gamma Function ........................................ 413
Weierstrass Product .................................... 413
The Gauss Multiplication Formula (Distribution
Relation) .............................................. 416
The (Other) Gauss Formula .............................. 418
The Mellin Transform ................................... 420
The Stirling Formula ................................... 422
Proof of Stirling's Formula ............................ 424
§3. The Lerch Formula ......................................... 431
§4. Zeta Functions ............................................ 433
CHAPTER XVI The Prime Number Theorem .......................... 440
§1. Basic Analytic Properties of the Zeta Function ............ 441
§2. The Main Lemma and its Application ........................ 446
§3. Proof of the Main Lemma ................................... 449
Appendix .................................................. 453
§1. Summation by Parts and Non-Absolute Convergence ........... 453
§2. Difference Equations ...................................... 457
§3. Analytic Differential Equations ........................... 461
§4. Fixed Points of a Fractional Linear Transformation ........ 465
§5. Cauchy's Formula for С∞ Functions ......................... 467
§6. Cauchy's Theorem for Locally Integrable Vector Fields ..... 472
§7. More on Cauchy-Riemann .................................... 477
Bibliography .................................................. 479
Index ......................................................... 481
|
