Function Theory of Several Complex Variables

Function Theory of Several Complex Variables
Title Function Theory of Several Complex Variables PDF eBook
Author Steven George Krantz
Publisher American Mathematical Soc.
Pages 586
Release 2001
Genre Mathematics
ISBN 0821827243

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Emphasizing integral formulas, the geometric theory of pseudoconvexity, estimates, partial differential equations, approximation theory, inner functions, invariant metrics, and mapping theory, this title is intended for the student with a background in real and complex variable theory, harmonic analysis, and differential equations.

Analytic Functions of Several Complex Variables

Analytic Functions of Several Complex Variables
Title Analytic Functions of Several Complex Variables PDF eBook
Author Robert Clifford Gunning
Publisher American Mathematical Soc.
Pages 338
Release 2009
Genre Mathematics
ISBN 0821821652

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The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. This title intends to provide an extensive introduction to the Oka-Cartan theory and some of its applications, and to the general theory of analytic spaces.

Methods of the Theory of Functions of Many Complex Variables

Methods of the Theory of Functions of Many Complex Variables
Title Methods of the Theory of Functions of Many Complex Variables PDF eBook
Author Vasiliy Sergeyevich Vladimirov
Publisher Courier Corporation
Pages 370
Release 2007-01-01
Genre Mathematics
ISBN 0486458121

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This systematic exposition outlines the fundamentals of the theory of single sheeted domains of holomorphy. It further illustrates applications to quantum field theory, the theory of functions, and differential equations with constant coefficients. Students of quantum field theory will find this text of particular value. The text begins with an introduction that defines the basic concepts and elementary propositions, along with the more salient facts from the theory of functions of real variables and the theory of generalized functions. Subsequent chapters address the theory of plurisubharmonic functions and pseudoconvex domains, along with characteristics of domains of holomorphy. These explorations are further examined in terms of four types of domains: multiple-circular, tubular, semitubular, and Hartogs' domains. Surveys of integral representations focus on the Martinelli-Bochner, Bergman-Weil, and Bochner representations. The final chapter is devoted to applications, particularly those involved in field theory. It employs the theory of generalized functions, along with the theory of functions of several complex variables.

Elementary Theory of Analytic Functions of One or Several Complex Variables

Elementary Theory of Analytic Functions of One or Several Complex Variables
Title Elementary Theory of Analytic Functions of One or Several Complex Variables PDF eBook
Author Henri Cartan
Publisher Courier Corporation
Pages 242
Release 2013-04-22
Genre Mathematics
ISBN 0486318672

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Basic treatment includes existence theorem for solutions of differential systems where data is analytic, holomorphic functions, Cauchy's integral, Taylor and Laurent expansions, more. Exercises. 1973 edition.

Geometric Function Theory in Several Complex Variables

Geometric Function Theory in Several Complex Variables
Title Geometric Function Theory in Several Complex Variables PDF eBook
Author Junjirō Noguchi
Publisher American Mathematical Soc.
Pages 292
Release 1990
Genre Mathematics
ISBN 9780821845332

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An English translation of a book that first appeared in Japanese. It provides an account of recent developments in geometric function theory in several complex variables and presents fundamental descriptions of positive currents, plurisubharmonic functions and meromorphic mappings.

Analytic Function Theory of Several Variables

Analytic Function Theory of Several Variables
Title Analytic Function Theory of Several Variables PDF eBook
Author Junjiro Noguchi
Publisher Springer
Pages 407
Release 2016-08-16
Genre Mathematics
ISBN 9811002916

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The purpose of this book is to present the classical analytic function theory of several variables as a standard subject in a course of mathematics after learning the elementary materials (sets, general topology, algebra, one complex variable). This includes the essential parts of Grauert–Remmert's two volumes, GL227(236) (Theory of Stein spaces) and GL265 (Coherent analytic sheaves) with a lowering of the level for novice graduate students (here, Grauert's direct image theorem is limited to the case of finite maps).The core of the theory is "Oka's Coherence", found and proved by Kiyoshi Oka. It is indispensable, not only in the study of complex analysis and complex geometry, but also in a large area of modern mathematics. In this book, just after an introductory chapter on holomorphic functions (Chap. 1), we prove Oka's First Coherence Theorem for holomorphic functions in Chap. 2. This defines a unique character of the book compared with other books on this subject, in which the notion of coherence appears much later.The present book, consisting of nine chapters, gives complete treatments of the following items: Coherence of sheaves of holomorphic functions (Chap. 2); Oka–Cartan's Fundamental Theorem (Chap. 4); Coherence of ideal sheaves of complex analytic subsets (Chap. 6); Coherence of the normalization sheaves of complex spaces (Chap. 6); Grauert's Finiteness Theorem (Chaps. 7, 8); Oka's Theorem for Riemann domains (Chap. 8). The theories of sheaf cohomology and domains of holomorphy are also presented (Chaps. 3, 5). Chapter 6 deals with the theory of complex analytic subsets. Chapter 8 is devoted to the applications of formerly obtained results, proving Cartan–Serre's Theorem and Kodaira's Embedding Theorem. In Chap. 9, we discuss the historical development of "Coherence".It is difficult to find a book at this level that treats all of the above subjects in a completely self-contained manner. In the present volume, a number of classical proofs are improved and simplified, so that the contents are easily accessible for beginning graduate students.

Holomorphic Functions and Integral Representations in Several Complex Variables

Holomorphic Functions and Integral Representations in Several Complex Variables
Title Holomorphic Functions and Integral Representations in Several Complex Variables PDF eBook
Author R. Michael Range
Publisher Springer Science & Business Media
Pages 405
Release 2013-03-09
Genre Mathematics
ISBN 1475719183

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The subject of this book is Complex Analysis in Several Variables. This text begins at an elementary level with standard local results, followed by a thorough discussion of the various fundamental concepts of "complex convexity" related to the remarkable extension properties of holomorphic functions in more than one variable. It then continues with a comprehensive introduction to integral representations, and concludes with complete proofs of substantial global results on domains of holomorphy and on strictly pseudoconvex domains inC", including, for example, C. Fefferman's famous Mapping Theorem. The most important new feature of this book is the systematic inclusion of many of the developments of the last 20 years which centered around integral representations and estimates for the Cauchy-Riemann equations. In particu lar, integral representations are the principal tool used to develop the global theory, in contrast to many earlier books on the subject which involved methods from commutative algebra and sheaf theory, and/or partial differ ential equations. I believe that this approach offers several advantages: (1) it uses the several variable version of tools familiar to the analyst in one complex variable, and therefore helps to bridge the often perceived gap between com plex analysis in one and in several variables; (2) it leads quite directly to deep global results without introducing a lot of new machinery; and (3) concrete integral representations lend themselves to estimations, therefore opening the door to applications not accessible by the earlier methods.