Functions of One Complex Variable II

This is the sequel to my book Functions of One Complex Variable I, and probably a good opportunity to express my appreciation to the mathemat ical community for its reception of that work. In retrospect, writing that book was a crazy venture. As a graduate student I had had one of the worst learning experiences of my career when I took complex analysis; a truly bad teacher. As a non-tenured assistant professor, the department allowed me to teach the graduate course in complex analysis. They thought I knew the material; I wanted to learn it. I adopted a standard text and shortly after beginning to prepare my lectures I became dissatisfied. All the books in print had virtues; but I was educated as a modern analyst, not a classical one, and they failed to satisfy me. This set a pattern for me in learning new mathematics after I had become a mathematician. Some topics I found satisfactorily treated in some sources; some I read in many books and then recast in my own style. There is also the matter of philosophy and point of view. Going from a certain mathematical vantage point to another is thought by many as being independent of the path; certainly true if your only objective is getting there. But getting there is often half the fun and often there is twice the value in the journey if the path is properly chosen.

Zusammenfassung
Addresses many problems on the theory of functions of one complex variable. This book treats several topics in geometric function theory and potential theory in the plane, such as conformal equivalence for simply connected regions, de Branges' proof of the Bieberbach conjecture, Hardy spaces on the disk, and potential theory in the plane.

Inhalt
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.