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.