Graduate Texts in Mathematics. 159. New York, NY: Springer-Verlag. xvi, 394 p. DM 88.00; öS 686.40; sFr 84.50 (1995).

“Functions of one complex variable” by John B. Conway appeared in (1973;

Zbl 0277.30001) as Number 11 in the Springer series “Graduate Texts in Mathematics”. It was deservedly well-received by the mathematical community. For a generation, “Conway” has been to graduate students in American universities what “Churchill” was for undergraduates. In 1973 no one, including the author, expected that there would be a second volume. In his preface to “Functions of one complex variable. II.”, he tells something of his motivation for writing it. “The topics in this volume are some of the parts of analytic function theory that I have found useful for my work in operator theory or enjoyable in themselves; usually both. Many also fall into the category of topics that I found difficult to dig out of the literature.” “Getting there” (i.e., from one mathematical topic to another) “is often half the fun, and often there is twice the value in the journey if the path is properly chosen.” Professor Conway has chosen his topics and paths well.
Volume II consists of Chapters 13 through 21. While there are quick reviews when appropriate, the student is assumed to have learned and be able to use introductory measure theory, topology, and functional analysis. This is especially true in the last four chapters. The author employs a comfortable writing style throughout. Every chapter and many sections start with a chatty discussion of what lies ahead. We are frequently cautioned against jumping to conclusions, and alerted to what are the points-at-issue. Sources of approaches to topics are acknowledged, and references cited where the researcher will find them convenient.
Some basic ideas are collected in Chapter 13. Chapters 14 and 15 concern conformal equivalence of simply connected and of finitely connected regions, with an emphasis on boundary values via the theory of prime ends. Chapter 16 is about analytic covering maps, the modular function, and the uniformization theorem. De Branges’ proof of the Bieberbach conjecture, following the approach of Fitzgerald and Pommerenke, makes up Chapter 17. Chapter 18 and 20 are relatively short. Topics from Chapter 18 include Bergman spaces, distributions, and the Cauchy transform. The Nevanlinna class and Hardy spaces on the disk are studied in Chapter 20. Chapters 19 and 21, on the other hand, give an ample treatment of harmonic functions and potential theory in the plane. Topics studied include Green functions, the Dirichlet Principle (via Sobolev spaces), harmonic measure, logarithmic capacity, and the finite topology. Students who make this trip through “Conway II” should find themselves ready for the research literature of complex analytic functions of a single variable.