Functions of One Complex Variable II

Description

Preface Contents 13 Return to Basics 1 Regions and Curves 2 Derivatives and Other Recollections 3 Harmonic Conjugates and Primitives 4 Analytic Arcs and the Reflection Principle 5 Boundary Values for Bounded Analytic Functions 14 Conformal Equivalence for Simply Connected Regions 1 Elementary Properties and Examples 2 Crosscuts 3 Prime Ends 4 Impressions of a Prime End 5 Boundary Values of Riemann Maps 6 The Area Theorem 7 Disk Mappings: The Class S 15 Conformal Equivalence for Finitely Connected Regions 1 Analysis on a Finitely Connected Region 2 Conformal Equivalence with an Analytic Jordan Region 3 Boundary Values for a Conformal Equivalence Between Finitely Connected Jordan Regions 4 Convergence of Univalent Functions 5 Conformal Equivalence with a Circularly Slit Annulus 6 Conformal Equivalence with a Circularly Slit Disk 7 Conformal Equivalence with a Circular Region 16 Analytic Covering Maps 1 Results for Abstract Covering Spaces 2 Analytic Covering Spaces 3 The Modular Function 4 Applications of the Modular Function 5 The Existence of the Universal Analytic Covering Map 17 De Branges's Proof of the Bieberbach Conjecture 1 Subordination 2 Loewner Chains 3 Loewner's Differential Equation 4 The Milin Conjecture 5 Some Special Functions 6 The Proof of de Branges's Theorem 18 Some Fundamental Concepts from Analysis 1 Bergman Spaces of Analytic and Harmonic Functions 2 Partitions of Unity 3 Convolution in Euclidean Space 4 Distributions 5 The Cauchy Transform 6 An Application: Rational Approximation 7 Fourier Series and Cesaro Sums 19 Harmonic Functions Redux 1 Harmonic Functions on the Disk 2 Fatou's Theorem 3 Semicontinuous Functions 4 Subharmonic Functions 5 The Logarithmic Potential 6 An Application: Approximation by Harmonic Functions 7 The Dirichlet Problem 8 Harmonic Majorants 9 The Green Function 10 Regular Points for the Dirichlet Problem 11 The Dirichlet Principle and Sobolev Spaces 20 Hardy Spaces on the Disk 1 Definitions and Elementary Properties 2 The Nevanlinna Class 3 Factorization of Functions in the Nevanlinna Class 4 The Disk Algebra 5 The Invariant Subspaces of H^p 6 Szego's Theorem 21 Potential Theory in the Plane 1 Harmonic Measure 2 The Sweep of a Measure 3 The Robin Constant 4 The Green Potential 5 Polar Sets 6 More on Regular Points 7 Logarithmic Capacity: Part 1 8 Some Applications and Examples of Logarithmic Capacity 9 Removable Singularities for Functions in the Bergman Space 10 Logarithmic Capacity: Part 2 11 The Transfinite Diameter and Logarithmic Capacity 12 The Refinement of a Subharmonic Function 13 The Fine Topology 14 Wiener's Criterion for Regular Points References List of Symbols Index