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 The present volume has five chapters. The first chapter deals with the fundamental properties of analytic funcrions in the space of several complex variables, and the second chapter with the properties of analytic funCtions in covering regions over a suitable space. These two chapters may be considered as a textbook for readers who are looking for basic information, in as elementary a form as possible, about the theory of functions of several complex variables. The next three chapters deal respectively with complex spaces, inten representations of functions of several complex variables, and functions meromorphic of one another in content but each of in the whole space. They are independent them makes a great deal of use of the material of the first two chapters. In contradistinction to the first two chapters, the lasr three are to a great extent in the nature of a survey. These chapters may serve as an introduction to the current technical literature on the various branches of the theory of functions. The actual exposition itself is preceded by an introductory essay giving the most frequently used information from closely related mathematical disciplines. It is recommended that the reader refer to this essay whenever he finds it necessary.   The present book constitutes the first part of a second edition, considerably revised and enlarged, of the author’s book Theory of analytic functions of several complex variables published in 1948. The second part, which is to appear soon after the first, will discuss a number of special chapters in the theory of functions. At the request of the author the firsr draft of the text of subsections 1—3, § 23, dealing with integral representations in n-circular regions, was written by L.A. Aizenberg, subsections 4—6, §23, dealing with integral representations in tubular regions, by S.G. Gindikin, and section 26, dealing with methods of characterizing the growth of entire functions, by L.I. Ronkin. These sections contain a number of new results, which are due to the above mentioned persons and are introduced here, as a rule, without reference to the original articles. An exposition of a number of original results referring to integral representations was kindly placed at my disposal by A.A. Temljakov. I am also indebted to L.A. Aizenberg and D.B. Fuks, who looked over the entire text while it was being prepared for the press and gave me valuable advice. To all the above persons I wish to express my profound gratitude. Many sections of this book were first presented to the seminar on the theory of analytic functions at the University of Moscow. I wish to take this opportunity of thanking the members of the seminar, and several other mathematicians, who looked over various parts of the book and sent me their suggestions.  Translated From : ВВЕДЕНИЕ В TEOPИЮ АНАЛИТИЧЕСКИХ ФУНК МНОГИХ КОМПЛЕКСНЫХ ПЕРЕМЕННЫХ - Б.А. ФУКС  Государственное Издательство Физике—Матсматхічсской Литературы Москва - 1962 VVEDENIYe V TEOPIYU ANALITICHESKIKH FUNK MNOGIKH KOMPLEKSNYKH PEREMENNYKH - B.A. FUKS  Gosudarstvennoye Izdatel'stvo Fizike—Matsmatkhíchsskoy Literatury Moskva - 1962

 The present volume has five chapters. The first chapter deals with the fundamental properties of analytic funcrions in the space of several complex variables, and the second chapter with the properties of analytic funCtions in covering regions over a suitable space. These two chapters may be considered as a textbook for readers who are looking for basic information, in as elementary a form as possible, about the theory of functions of several complex variables. The next three chapters deal respectively with complex spaces, inten representations of functions of several complex variables, and functions meromorphic of one another in content but each of in the whole space. They are independent them makes a great deal of use of the material of the first two chapters. In contradistinction to the first two chapters, the lasr three are to a great extent in the nature of a survey. These chapters may serve as an introduction to the current technical literature on the various branches of the theory of functions. The actual exposition itself is preceded by an introductory essay giving the most frequently used information from closely related mathematical disciplines. It is recommended that the reader refer to this essay whenever he finds it necessary.   The present book constitutes the first part of a second edition, considerably revised and enlarged, of the author’s book Theory of analytic functions of several complex variables published in 1948. The second part, which is to appear soon after the first, will discuss a number of special chapters in the theory of functions. At the request of the author the firsr draft of the text of subsections 1—3, § 23, dealing with integral representations in n-circular regions, was written by L.A. Aizenberg, subsections 4—6, §23, dealing with integral representations in tubular regions, by S.G. Gindikin, and section 26, dealing with methods of characterizing the growth of entire functions, by L.I. Ronkin. These sections contain a number of new results, which are due to the above mentioned persons and are introduced here, as a rule, without reference to the original articles. An exposition of a number of original results referring to integral representations was kindly placed at my disposal by A.A. Temljakov. I am also indebted to L.A. Aizenberg and D.B. Fuks, who looked over the entire text while it was being prepared for the press and gave me valuable advice. To all the above persons I wish to express my profound gratitude. Many sections of this book were first presented to the seminar on the theory of analytic functions at the University of Moscow. I wish to take this opportunity of thanking the members of the seminar, and several other mathematicians, who looked over various parts of the book and sent me their suggestions.  Translated From : ВВЕДЕНИЕ В TEOPИЮ АНАЛИТИЧЕСКИХ ФУНК МНОГИХ КОМПЛЕКСНЫХ ПЕРЕМЕННЫХ - Б.А. ФУКС  Государственное Издательство Физике—Матсматхічсской Литературы Москва - 1962 VVEDENIYe V TEOPIYU ANALITICHESKIKH FUNK MNOGIKH KOMPLEKSNYKH PEREMENNYKH - B.A. FUKS  Gosudarstvennoye Izdatel'stvo Fizike—Matsmatkhíchsskoy Literatury Moskva - 1962

 The present volume is closely related in its contents to the author’s book Theory of analytic functions of several complex variables, published in English translation by the American Mathematical Society in 1963. These two volumes together constitute the second edition, considerably revised and enlarged, of the monograph Theory of analytic functions of several complex variables published in 1948. In the second edition, as well as in the first, the author does not aim to cover, even to any extent, all the material which has accumulated in the theory of analytic functions of several complex variables.   The present volume is divided into five chapters: approximation of functions and domains, coherent analytic sheaves and the solution of fundamental problems, domains analytically convex in the "sense of Hartogs, holomorphic extension of domains, biholomorphic mappings. In order to understand these chapters, the reader should be familiar with the concepts contained in the first two chapters of Theory of analytic functions of several complex variables. In addition, for § 2 of Chapter II the reader must be acquainted with Weil’s integral representations (§ 22, Chapter IV, (1)) and Cousin’s first theorem (§ 25, Chapter V, (1)), for §§8—12 of Chapter II with properties of holomorphically complete complex manifolds (§ 14.1 and §18. 3—4, Chapter III, (1)), for §l4 of Chapter III with Weil’s integral representations and the methods of solving Cousin’s first problem (throughout §25, Chapter V, (I) and §7, Chapter :11) and with properties of sequences of domains (§ 6, Chapter I), for §§ 17 and 18 of Chapter IV with the theory of plurisubharmonic functions (§13, Chapter III) and for Chapter V with properties of the kernel function in a domain (§§ 4 and 5, Chapter 1). The remaining parts Of the present volume, except for certain cross-references, are independent of one another and of Chapters III, IV and V of the first part of the book. To shorten the volume and simplify the text the proofs of several propositions are not developed in the most general form. For example, Theorems (-A) and (B) of H. Cartan are proved for complex manifolds, but not for spaces; the theorem of K. Oka on the domains convex in the sense of Hartogs is proved for the case of spaces of only two complex variables. The "edge of the wedge” theorem of N.N. Bogoljubov is also proved in a simplified form by introducing some hypotheses. The actual text of the book itself is preceded by an introductory essay giving the most frequently used information from closely related mathematical disciplines. It is recommended that the reader refer to this essay whenever he finds it necessary.   At the request of the author, the first draft of the text of §§8—12, dealing with the theory of coherent analytic sheaves and its application to the solution of the fundamental problem, was written by D.B. Fuks, §19, dealing with the "edge of the wedge” theorem, by V.S. Vladimirov, and §24, dealing with homogeneous bounded domains, by S.G. Gindikin. The latter two sections contain new results which are due to the above-mentioned persons and are introduced here, as a rule, without reference to the original articles. The author is indebted for advice and a number of valuable remarks to L.A. Ai’zenberg, who looked over the entire text while it was being prepared for the press, and to V.S. Vladimirov, who looked over the text of Chapters III and V. To all the above persons I wish to express my profound gratitude. I also wish to take this opportunity of thanking other mathematicians who looked over various parts of the book and sent me their suggestions. Results belonging to many mathematicians are presented in this book. It should be noted, however, that the greatest influence on its contents is due to works of S. Bergman, concerning the kernel function in a domain and its applications, of G. Bremermann, concerning domains convex in the sense of Hartogs and the holomorphic extension. of domains, of H. Cartan who established, in collaboration with J.P. Sam and others, the theory of coherent analytic sheaves and its applications to many important problems of the theory of functions, and of K. Oka, concerning approximation of functions, Cousin’s problems and the solution of the inverse problem of Hartogs . Translated from : СПЕЦИАЛЬНЫЕ ГЛАВЫ ТЕОРИИ АНАЛИТИЧЕСКИХ ФУНКЦИИ МНОГИХ.КОМПЛЕКСНЫХ ПЕРЕМЕННЫХ  Б.А. ФУКС  Государственное Издательство Физике—Математической Литературы  Москва 1963 

 The present volume is closely related in its contents to the author’s book Theory of analytic functions of several complex variables, published in English translation by the American Mathematical Society in 1963. These two volumes together constitute the second edition, considerably revised and enlarged, of the monograph Theory of analytic functions of several complex variables published in 1948. In the second edition, as well as in the first, the author does not aim to cover, even to any extent, all the material which has accumulated in the theory of analytic functions of several complex variables.   The present volume is divided into five chapters: approximation of functions and domains, coherent analytic sheaves and the solution of fundamental problems, domains analytically convex in the "sense of Hartogs, holomorphic extension of domains, biholomorphic mappings. In order to understand these chapters, the reader should be familiar with the concepts contained in the first two chapters of Theory of analytic functions of several complex variables. In addition, for § 2 of Chapter II the reader must be acquainted with Weil’s integral representations (§ 22, Chapter IV, (1)) and Cousin’s first theorem (§ 25, Chapter V, (1)), for §§8—12 of Chapter II with properties of holomorphically complete complex manifolds (§ 14.1 and §18. 3—4, Chapter III, (1)), for §l4 of Chapter III with Weil’s integral representations and the methods of solving Cousin’s first problem (throughout §25, Chapter V, (I) and §7, Chapter :11) and with properties of sequences of domains (§ 6, Chapter I), for §§ 17 and 18 of Chapter IV with the theory of plurisubharmonic functions (§13, Chapter III) and for Chapter V with properties of the kernel function in a domain (§§ 4 and 5, Chapter 1). The remaining parts Of the present volume, except for certain cross-references, are independent of one another and of Chapters III, IV and V of the first part of the book. To shorten the volume and simplify the text the proofs of several propositions are not developed in the most general form. For example, Theorems (-A) and (B) of H. Cartan are proved for complex manifolds, but not for spaces; the theorem of K. Oka on the domains convex in the sense of Hartogs is proved for the case of spaces of only two complex variables. The "edge of the wedge” theorem of N.N. Bogoljubov is also proved in a simplified form by introducing some hypotheses. The actual text of the book itself is preceded by an introductory essay giving the most frequently used information from closely related mathematical disciplines. It is recommended that the reader refer to this essay whenever he finds it necessary.   At the request of the author, the first draft of the text of §§8—12, dealing with the theory of coherent analytic sheaves and its application to the solution of the fundamental problem, was written by D.B. Fuks, §19, dealing with the "edge of the wedge” theorem, by V.S. Vladimirov, and §24, dealing with homogeneous bounded domains, by S.G. Gindikin. The latter two sections contain new results which are due to the above-mentioned persons and are introduced here, as a rule, without reference to the original articles. The author is indebted for advice and a number of valuable remarks to L.A. Ai’zenberg, who looked over the entire text while it was being prepared for the press, and to V.S. Vladimirov, who looked over the text of Chapters III and V. To all the above persons I wish to express my profound gratitude. I also wish to take this opportunity of thanking other mathematicians who looked over various parts of the book and sent me their suggestions. Results belonging to many mathematicians are presented in this book. It should be noted, however, that the greatest influence on its contents is due to works of S. Bergman, concerning the kernel function in a domain and its applications, of G. Bremermann, concerning domains convex in the sense of Hartogs and the holomorphic extension. of domains, of H. Cartan who established, in collaboration with J.P. Sam and others, the theory of coherent analytic sheaves and its applications to many important problems of the theory of functions, and of K. Oka, concerning approximation of functions, Cousin’s problems and the solution of the inverse problem of Hartogs . Translated from : СПЕЦИАЛЬНЫЕ ГЛАВЫ ТЕОРИИ АНАЛИТИЧЕСКИХ ФУНКЦИИ МНОГИХ.КОМПЛЕКСНЫХ ПЕРЕМЕННЫХ  Б.А. ФУКС  Государственное Издательство Физике—Математической Литературы  Москва 1963 

 The present volume is closely related in its contents to the author’s book Theory of analytic functions of several complex variables, published in English translation by the American Mathematical Society in 1963. These two volumes together constitute the second edition, considerably revised and enlarged, of the monograph Theory of analytic functions of several complex variables published in 1948. In the second edition, as well as in the first, the author does not aim to cover, even to any extent, all the material which has accumulated in the theory of analytic functions of several complex variables.   The present volume is divided into five chapters: approximation of functions and domains, coherent analytic sheaves and the solution of fundamental problems, domains analytically convex in the "sense of Hartogs, holomorphic extension of domains, biholomorphic mappings. In order to understand these chapters, the reader should be familiar with the concepts contained in the first two chapters of Theory of analytic functions of several complex variables. In addition, for § 2 of Chapter II the reader must be acquainted with Weil’s integral representations (§ 22, Chapter IV, (1)) and Cousin’s first theorem (§ 25, Chapter V, (1)), for §§8—12 of Chapter II with properties of holomorphically complete complex manifolds (§ 14.1 and §18. 3—4, Chapter III, (1)), for §l4 of Chapter III with Weil’s integral representations and the methods of solving Cousin’s first problem (throughout §25, Chapter V, (I) and §7, Chapter :11) and with properties of sequences of domains (§ 6, Chapter I), for §§ 17 and 18 of Chapter IV with the theory of plurisubharmonic functions (§13, Chapter III) and for Chapter V with properties of the kernel function in a domain (§§ 4 and 5, Chapter 1). The remaining parts Of the present volume, except for certain cross-references, are independent of one another and of Chapters III, IV and V of the first part of the book. To shorten the volume and simplify the text the proofs of several propositions are not developed in the most general form. For example, Theorems (-A) and (B) of H. Cartan are proved for complex manifolds, but not for spaces; the theorem of K. Oka on the domains convex in the sense of Hartogs is proved for the case of spaces of only two complex variables. The "edge of the wedge” theorem of N.N. Bogoljubov is also proved in a simplified form by introducing some hypotheses. The actual text of the book itself is preceded by an introductory essay giving the most frequently used information from closely related mathematical disciplines. It is recommended that the reader refer to this essay whenever he finds it necessary.   At the request of the author, the first draft of the text of §§8—12, dealing with the theory of coherent analytic sheaves and its application to the solution of the fundamental problem, was written by D.B. Fuks, §19, dealing with the "edge of the wedge” theorem, by V.S. Vladimirov, and §24, dealing with homogeneous bounded domains, by S.G. Gindikin. The latter two sections contain new results which are due to the above-mentioned persons and are introduced here, as a rule, without reference to the original articles. The author is indebted for advice and a number of valuable remarks to L.A. Ai’zenberg, who looked over the entire text while it was being prepared for the press, and to V.S. Vladimirov, who looked over the text of Chapters III and V. To all the above persons I wish to express my profound gratitude. I also wish to take this opportunity of thanking other mathematicians who looked over various parts of the book and sent me their suggestions. Results belonging to many mathematicians are presented in this book. It should be noted, however, that the greatest influence on its contents is due to works of S. Bergman, concerning the kernel function in a domain and its applications, of G. Bremermann, concerning domains convex in the sense of Hartogs and the holomorphic extension. of domains, of H. Cartan who established, in collaboration with J.P. Sam and others, the theory of coherent analytic sheaves and its applications to many important problems of the theory of functions, and of K. Oka, concerning approximation of functions, Cousin’s problems and the solution of the inverse problem of Hartogs . Translated from : СПЕЦИАЛЬНЫЕ ГЛАВЫ ТЕОРИИ АНАЛИТИЧЕСКИХ ФУНКЦИИ МНОГИХ.КОМПЛЕКСНЫХ ПЕРЕМЕННЫХ  Б.А. ФУКС  Государственное Издательство Физике—Математической Литературы  Москва 1963