This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable. It treats several topics in geometric function theory as well as potential theory in the plane. In particular it covers: conformal equivalence for simply connected regions, conformal equivalence for finitely connected regions, analytic covering maps, de Branges' proof of the Bieberbach conjecture, harmonic functions, Hardy spaces on the disk, potential theory in the plane. The level of the material is gauged for graduate students. Chapters XIII through XVII have the same prerequisites as the first volume of this text, GTM 11. For the remainder of the text it is assumed that the reader has a knowledge of integration theory and functional analysis. Definitions and theorems are stated clearly and precisely. Also contained in this book is an abundance of exercises of various degrees of difficulty.
Author(s): John B. Conway
Series: Graduate Texts in Mathematics Pt. 2
Publisher: Springer
Year: 1995
Language: English
Pages: 412
Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface......Page 6
Contents of Volume II......Page 10
Contents of Volume I......Page 14
1 Regions and Curves ......Page 16
2 Derivatives and Other Recollections ......Page 21
3 Harmonic Conjugates and Primitives ......Page 29
4 Analytic Arcs and the Reflection Principle ......Page 31
5 Boundary Values for Bounded Analytic Functions ......Page 36
1 Elementary Properties and Examples ......Page 44
2 Croascuts ......Page 48
3 Prime Ends ......Page 55
4 Impressions of a Prime End ......Page 60
5 Boundary Values of Riemann Maps ......Page 63
6 The Area Theorem ......Page 71
7 Disk Mappings: The Class S ......Page 76
1 Analysis on a Finitely Connected Region ......Page 86
2 Conformal Equivalence with an Analytic Jordan Region ......Page 91
3 Boundary Values for a Conformal Equivalence Between Finitely Connected Jordan Regions ......Page 96
4 Convergence of Univalent Functions ......Page 100
5 Conformal Equivalence with a Circularly Slit Annulus ......Page 105
6 Conformal Equivalence with a Circularly Slit Disk ......Page 112
7 Conformal Equivalence with a Circular Region ......Page 115
1 Results for Abstract Covering Spaces ......Page 124
2 Analytic Covering Spaces ......Page 128
3 The Modular Function ......Page 131
4 Applications of the Modular Function ......Page 138
5 The Existence of the Universal Analytic Covering Map ......Page 140
1 Subordination ......Page 148
2 Loewner Chains ......Page 151
3 Loewner's Differential Equation ......Page 157
4 The Mum Conjecture ......Page 163
5 Some Special Functions ......Page 171
6 The Proof of de Branges's Theorem ......Page 175
1 Bergman Spaces of Analytic and Harmonic Functions ......Page 184
2 Partitions of Unity ......Page 189
3 Convolution in Euclidean Space ......Page 192
4 Distributions ......Page 200
5 The Cauchy Transform ......Page 207
6 An Application: Rational Approximation ......Page 211
7 Fourier Series and Ces?o Sums ......Page 213
1 Harmonic Functions on the Disk ......Page 220
2 Fatou's Theorem ......Page 225
3 Semicontinuous Functions ......Page 232
4 Subharmonic Functions ......Page 235
5 The Logarithmic Potential ......Page 244
6 An Application: Approximation by Harmonic Functions ......Page 250
7 The Dirichiet Problem ......Page 252
8 Harmonic Majorants ......Page 260
9 The Green Function ......Page 261
10 Regular Points for the Dirichiet Problem ......Page 268
11 The Dirichiet Principle and Sobolev Spaces ......Page 274
1 Definitions and Elementary Properties ......Page 284
2 The Nevanlinna Class ......Page 287
3 Factorization of Functions in the Nevanlinna Class ......Page 293
4 The Disk Algebra ......Page 301
5 The Invariant Subspaces of flP ......Page 305
6 Szego's Theorem ......Page 310
1 Harmonic Measure ......Page 316
2 The Sweep of a Measure ......Page 326
3 The Robin Constant ......Page 328
4 The Green Potential ......Page 330
5 Polar Sets ......Page 335
6 More on Regular Points ......Page 343
7 Logarithmic Capacity: Part 1 ......Page 346
8 Some Applications and Examples of Logarithmic Capacity ......Page 354
9 Removable Singularities for Functions in the Bergman Space ......Page 359
10 Logarithmic Capacity: Part 2 ......Page 367
11 The Transfinite Diameter and Logarithmic Capacity ......Page 370
12 The Refinement of a Subharmonic Function ......Page 375
13 The Fine Topology ......Page 382
14 Wiener's criterion for Regular Points ......Page 391
References ......Page 399
List of Symbols ......Page 404
Index ......Page 406
Back Cover......Page 412
