Complex Function Theory (eBook)

Front Cover 1
Complex Function Theory 6
Copyright Page 7
Contents 12
Preface 8
PART ONE 18
Chapter I. The Real Field 20
1. Sets 20
2. Groups, Rings, Fields 22
3. Ordered Fields 24
4. The Real Number System 26
5. Real Sequences, Monotone Sequences 29
6. Limit Superior, Limit Inferior 32
7. Cauchy Sequences 33
Chapter II. The Complex Field. Limits 34
1. The Complex Field 35
2. Geometry of the Complex Plane 38
3. Limits 41
4. Infinite Series 44
5. Power Series 46
Chapter III. Topological and Metric Spaces 55
1. The Notion of a Topological Space 56
2. Compactness 60
3. Connectedness 63
4. Arc 66
5. Metric Space 67
6. Compactification 70
Chapter IV. Complex Differential Analysis 74
1. Complex-Valued Functions of One Real Variable 75
2. Complex-Valued Functions of One Complex Variable 79
3. Power Series 80
4. Further Properties and Applications of the Exponential Function, Trigonometric Functions 86
5. Fourier Aspects of Power Series 92
6. Functions of Two Real Variables 96
7. The Cauchy-Riemann Equation(s) 102
8. Harmonic Functions 104
9. The Theorem of Gauss on the Zeros of the Derivative of a Polynomial 108
Chapter V. Cauchy Theory 111
1. Line Integrals 112
2. The Simplicia1 Cauchy-Goursat Theorem 121
3. The Cauchy-Goursat Theorem for a Singular Cell and Some Consequences 126
4. Simple Connectivity, Primitives 141
5 . Analytic and Continuous Logarithms and Powers 144
6. Certain Integrals Representing Analytic Functions 148
7. Some Sufficient Conditions for Analyticity 156
8. Runge’s Theorem 157
9. Mapping Properties of Analytic Functions 160
Chapter VI. Laurent Expansion. Meromorphic Functions 167
1. Laurent Expansion 168
2. Isolated Singularities of Analytic Functions 175
3. Residue 181
4. Meromorphic Functions 184
Chapter VII. Further Applications of the Cauchy Theory 199
1. The Residue Theorem 200
2. Application of the Residue Theorem to the Calculation of Integrals 203
3. Application of the Method of Cauchy 208
Chapter VIII. The Zeros and Poles of Meromorphic Functions 211
1. The Theorems of Mittag-Leffler and Weierstrass 213
2. The Isomorphism Theorem of Bers 217
3. Infinite Products 221
4. Sums and Products 227
5. The Jensen Formula 230
6. Entire Functions of Finite Order 233
7. Bounded Analytic Functions 241
Chapter IX. The Gamma and Zeta Functions. Prime Number Theorem 246
1. The Gamma Function 247
2. The Riemann Zeta Function and Its Functional Equation 254
3. The Theorem of Hadamard and de la Vallée Poussin on the Zeros of the Riemann Zeta Function 259
4. The Prime Number Theorem of Hadamard and de la Vallée Poussin 260
PART TWO 268
Chapter X. Supplementary Developments Concerning C 270
1. Other Constructions of C 270
2. Characterizations of Absolute Value 271
3. Geometry in C 271
Chapter XI. Complex Differential Coefficients 277
1. Complex Differential Coefficients 277
2. Conformality 279
3. Complex Differential Coefficients and Harmonic Functions 281
Chapter XII. Topics in the Theory of Power Series 283
1. Formal Aspects 284
2. Sums, Cauchy Multiplication 285
3. Holomorphic Functions of Two Complex Variables 286
4. Cauchy’s Majorant Calculus 290
5. Vector-Valued Functions 295
6. Analytic Continuation 303
Chapter XIII. Harmonic and Subharmonic Functions 311
1. Poisson Integral and Applications 312
2. Weierstrass Approximation Theorem and the Dirichlet Problem for an Annulus 317
3. Principles of Schwarz and Carleman 319
4. Subharmonic Functions 321
Chapter XIV. Complements to the Cauchy Theory 327
1. Gaussian Sums 328
2. Doubly Periodic Functions 330
3. The Weierstrass Preparation Theorem 331
4. Runge’s Theorem on Polynomial Approximation 334
5. Integrals of Cauchy Type, Cauchy Principal Value 336
6. The Cauchy Integral Theorem for Weierstrass Classes 338
Chapter XV. Möbius Transformations 342
1. Classification of Möbius Transformations 342
Chapter XVI. The Modular Function . The Picard Theorems 346
1. The Modular Function . 347
2. The Big Picard Theorem 355
Chapter XVll. Some Questions Concerning Univalent Analytic Functions 357
1. The Riemann Mapping Theorem (plane Case) 358
2. Carathéodory’s Theorem on Variable Regions 360
3. Application of the Schwarz Reflexion Principle 362
4. Schwarz-Christoffel Functions 364
5. Regions with Simple Polygonal Frontiers 368
Chapter XVlll. Riemann Surfaces 371
1. Riemann Surfaces in the Sense of Weyl-Radó 372
2. Weierstrass Classes, Analytic Entities 375
3. Compact Analytic Entities and Riemann Surfaces 385
4. The Problem of Riemann and Klein 396
5. Distinguishing Meromorphic Functions 397
6. Existence Theorem for Auxiliary Harmonic Functions 400
7. Field Isomorphism and Conformal Equivalence of Compact Riemann Surfaces 406
References 411
Notation 417
Index 424