Introduction to complex analysis. Part 2: Functions of several variables. Textbook. (Vvedenie v kompleksnyj analiz. Chast’ 2: Funktsii neskol’kikh peremennykh. Uchebnoe posobie). 3rd ed., rev. and compl. (Russian) Zbl 0578.32001

Shabat, B. V.

Moskva: ”Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury. 464 p. R. 1.20 (1985).

This second volume, compared to the corresponding one of the preceding edition (1976) [for the first ed. see (1969; Zbl 0188.379); for part 1: Functions of one variable, 3rd ed. see (1985; Zbl 0574.30001) and 1st. ed. (1969; Zbl 0188.379), respectively] is completely transformed as a consequence of the impetuous development of the function theory of several complex variables in last decade.
The first chapter is devoted to holomorphic functions and maps of several complex variables. The second chapter (”Basic geometrical concepts”) presents integration theory on complex manifolds. In chapter 4, the author considers meromorphic functions, Cousin’s problem and residues. Chapter 5 (”Some problems of geometric theory”) contains a paragraph on invariant metrics (Bergman, Carathéodory and Kobayashi metrics), another one on the hyperbolic manifolds and the last on boundary behaviour of holomorphic functions in strictly pseudoconvex domains.
There is also a supplement on complex potential theory, where the author gives sketchy considerations on the generalization of harmonic functions in \({\mathbb{C}}^ n.\)
The book is provided also with a subject index. At the end of each chapter, there is a paragraph containing problems.

Reviewer: P.Caraman

MSC:

32-02	Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces
32A10	Holomorphic functions of several complex variables
32H99	Holomorphic mappings and correspondences
32C30	Integration on analytic sets and spaces, currents
32Sxx	Complex singularities
32A27	Residues for several complex variables
32A20	Meromorphic functions of several complex variables
32F45	Invariant metrics and pseudodistances in several complex variables
32E35	Global boundary behavior of holomorphic functions of several complex variables
32Q45	Hyperbolic and Kobayashi hyperbolic manifolds
31C05	Harmonic, subharmonic, superharmonic functions on other spaces
32T99	Pseudoconvex domains

Keywords:

analytic continuation; domain of holomorphy; holomorphic maps; Bergman metric; Carathéodory metric; function theory of several complex variables; holomorphic functions; integration theory on complex manifolds; meromorphic functions; Cousin’s problem; residues; invariant metrics; Kobayashi metrics; hyperbolic manifolds; boundary behaviour; strictly pseudoconvex domains; harmonic functions