单复变函数 第2卷 英文PDF电子书下载

13 Return to Basics 1

1 Regions and Curves 1

2 Derivatives and Other Recollections 6

3 Harmonic Conjugates and Primitives 14

4 Analytic Arcs and the Reflection Principle 16

5 Boundary Values for Bounded Analytic Functions 21

14 Conformal Equivalence for Simply Connected Regions 29

1 Elementary Properties and Examples 29

2 Crosscuts 33

3 Prime Ends 40

4 Impressions of a Prime End 45

5 Boundary Values of Riemann Maps 48

6 The Area Theorem 56

7 Disk Mappings: The Class S 61

15 Conformal Equivalence for Finitely Connected Regions 71

1 Analysis on a Finitely Connected Region 71

2 Conformal Equivalence with an Analytic Jordan Region 76

3 Boundary Values for a Conformal Equivalence Between Finitely Connected Jordan Regions 81

4 Convergence of Univalent Functions 85

5 Conformal Equivalence with a Circularly Slit Annulus 90

6 Conformal Equivalence with a Circularly Slit Disk 97

7 Conformal Equivalence with a Circular Region 100

16 Analytic Covering Maps 109

1 Results for Abstract Covering Spaces 109

2 Analytic Covering Spaces 113

3 The Modular Function 116

4 Applications of the Modular Function 123

5 The Existence of the Universal Analytic Covering Map 125

17 De Branges's Proof of the Bieberbach Conjecture 133

1 Subordination 133

2 Loewner Chains 136

3 Loewner's Differential Equation 142

4 The Milin Conjecture 148

5 Some Special Functions 156

6 The Proof of de Branges's Theorem 160

18 Some Fundamental Concepts from Analysis 169

1 Bergman Spaces of Analytic and Harmonic Functions 169

2 Partitions of Unity 174

3 Convolution in Euclidean Space 177

4 Distributions 185

5 The Cauchy Tr ansform 192

6 An Application: Rational Approximation 196

7 Fourier Series and Cesàro Sums 198

19 Harmonic Functions Redux 205

1 Harmonic Functions on the Disk 205

2 Fatou's Theorem 210

3 Semicontinuous Functions 217

4 Subharmonic Functions 220

5 The Logarithmic Potential 229

6 An Application: Approximation by Harmonic Functions 235

7 The Dirichlet Problem 237

8 Harmonic Majorants 245

9 The Green Function 246

10 Regular Points for the Dirichlet Problem 253

11 The Dirichlet Principle and Sobolev Spaces 259

20 Hardy Spaces on the Disk 269

1 Definitions and Elementa Properties 269

2 The Nevanlinna Class 272

3 Factorization of Functions in the Nevanlinna Class 278

4 The Disk Algebra 286

5 The Invariant Subspaces of Hp 290

6 Szeg？'s Theorem 295

21 Potential Theory in the Plane 301

1 Harmonic Measure 301

2 The Sweep of a Measure 311

3 The Robin Constant 313

4 The Green Potential 315

5 Polar Sets 320

6 More on Regular Points 328

7 Logarithmic Capacity: Part 1 331

8 Some Applications and Examples of Logarithmic Capacity 339

9 Removable Singularities for Functions in the Bergman Space 344

10 Logarithmic Capacity:Part 2 352

11 The Transfinite Diameter and Logarithmic Capacity 355

12 The Refinement of a Subharmonic Function 360

13 The Fine Topology 367

14 Wiener's criterion for Regular Points 376