Functions of One Complex Variable II

of Volume II. - 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 Cesàro 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 Hp. - 6 Szegö'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. Language: English