Introduction to Classical Complex Analysis (eBook)

Front Cover 1
An Introduction to Classical Complex Analysis 4
Copyright Page 5
Contents 6
Preface 10
Chapter 0. Prerequisites and Preliminaries 14
§ 1 Set Theory 14
§ 2 Algebra 15
§ 3 The Battlefield 15
§ 4 Metric Spaces 16
§ 5 Limsup and All That 19
§ 6 Continuous Functions 21
§ 7 Calculus 22
Chapter I. Curves, Connectedness and Convexity 23
§ 1 Elementary Results on Connectedness 23
§ 2 Connectedness of Intervals, Curves and Convex Sets 24
§ 3 The Basic Connectedness Lemma 29
§ 4 Components and Compact Exhaustions 30
§ 5 Connectivity of a Set 34
§ 6 Extension Theorems 38
Notes to Chapter I 40
Chapter II. (Complex) Derivative and (Curvilinear) Integrals 42
§ 1 Holomorphic and Harmonic Functions 42
§ 2 Integrals along Curves 45
§ 3 Differentiating under the Integral 48
§ 4 A Useful Sufficient Condition for Differentiability 50
Notes to Chapter II 51
Chapter III. Power Series and the Exponential Function 54
§ 1 Introduction 54
§ 2 Power Series 54
§ 3 The Complex Exponential Function 61
§ 4 Bernoulli Polynomials, Numbers and Functions 74
§ 5 Cauchy's Theorem Adumbrated 78
§ 6 Holomorphic Logarithms Previewed 79
Notes to Chapter III 81
Chapter IV. The Index and Some Plane Topology 84
§ 1 Introduction 84
§ 2 Curves Winding around Points 84
§ 3 Homotopy and the Index 91
§ 4 Existence of Continuous Logarithms 93
§ 5 The Jordan Curve Theorem 103
§ 6 Applications of the Foregoing Technology 107
§ 7 Continuous and Holomorphic Logarithms in Open Sets 112
§ 8 Simple Connectivity for Open Sets 114
Notes to Chapter IV 116
Chapter V. Consequences of the Cauchy–Goursat Theorem—Maximum Principles and the Local Theory 121
§ 1 Goursat’s Lemma and Cauchy’s Theorem for Starlike Regions 121
§ 2 Maximum Principles 128
§ 3 The Dirichlet Problem for Disks 135
§ 4 Existence of Power Series Expansions 145
§ 5 Harmonic Majorization 152
§ 6 Uniqueness Theorems 166
§ 7 Local Theory 173
Notes to Chapter V 184
Chapter VI. Schwarz' Lemma and its Many Applications 192
§ 1 Schwarz’ Lemma and the Conformal Automorphisms of Disks 192
§ 2 Many-to-one Maps of Disks onto Disks 198
§ 3 Applications to Half-planes, Strips and Annuli 199
§ 4 The Theorem of Carathéodory, Julia, Wolff, et al. 204
§ 5 Subordination 208
Notes to Chapter VI 216
Chapter VII. Convergent Sequences of Holomorphic Functions 219
§ 1 Convergence in H(U) 219
§ 2 Applications of the Convergence Theorems Boundedness Criteria
§ 3 Prescribing Zeros 238
§ 4 Elementary Iteration Theory 243
Notes to Chapter VII 252
Chapter VIII. Polynomial and Rational Approximation—Runge Theory 257
§ 1 The Basic Integral Representation Theorem 257
§ 2 Applications to Approximation 261
§ 3 Other Applications of the Integral Representation 266
§ 4 Some Special Kinds of Approximation 269
§ 5 Carleman’s Approximation Theorem 274
§ 6 Harmonic Functions in a Half-plane 277
Notes to Chapter VIII 290
Chapter IX. The Riemann Mapping Theorem 294
§ 1 Introduction 294
§ 2 The Proof of Carathéodory and Koebe 299
§ 3 Fejér and Riesz’ Proof 304
§ 4 Boundary Behavior for Jordan Regions 304
§ 5 A Few Applications of the Osgood–Taylor–Carathéodory Theorem 311
§ 6 More on Jordan Regions and Boundary Behavior 316
§ 7 Harmonic Functions and the General Dirichlet Problem 323
§ 8 The Dirichlet Problem and the Riemann Mapping Theorem 334
Notes to Chapter IX 338
Chapter X. Simple and Double Connectivity 345
§ 1 Simple Connectivity 345
§ 2 Double Connectivity 349
Notes to Chapter X 356
Chapter XI. Isolated Singularities 360
§ 1 Laurent Series and Classification of Singularities 360
§ 2 Rational Functions 367
§ 3 Isolated Singularities on the Circle of Convergence 376
§ 4 The Residue Theorem and Some Applications 378
§ 5 Specifying Principal Parts—Mittag-Leffler’s Theorem 391
§ 6 Meromorphic Functions 396
§ 7 Poisson’s Formula in an Annulus and Isolated Singularities of Harmonic Functions 399
Notes to Chapter XI 407
Chapter XII. Omitted Values and Normal Families 412
§ 1 Logarithmic Means and Jensen’s Inequality 412
§ 2 Miranda’s Theorem 418
§ 3 Immediate Applications of Miranda 433
§ 4 Normal Families and Julia’s Extension of Picard’s Great Theorem 437
§ 5 Sectorial Limit Theorems 442
§ 6 Applications to Iteration Theory 451
§ 7 Ostrowski’s Proof of Schottky’s Theorem 452
Notes to Chapter XII 457
Bibliography 463
Name Index 545
Subject Index 555
Symbol Index 569
Series Summed 570
Integrals Evaluated 571