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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
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