Several Complex Variables

Several Complex Variables
Author: H. Grauert
Publisher: Springer Science & Business Media
Total Pages: 213
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461298741

The present book grew out of introductory lectures on the theory offunctions of several variables. Its intent is to make the reader familiar, by the discussion of examples and special cases, with the most important branches and methods of this theory, among them, e.g., the problems of holomorphic continuation, the algebraic treatment of power series, sheaf and cohomology theory, and the real methods which stem from elliptic partial differential equations. In the first chapter we begin with the definition of holomorphic functions of several variables, their representation by the Cauchy integral, and their power series expansion on Reinhardt domains. It turns out that, in l:ontrast ~ 2 there exist domains G, G c en to the theory of a single variable, for n with G c G and G "# G such that each function holomorphic in G has a continuation on G. Domains G for which such a G does not exist are called domains of holomorphy. In Chapter 2 we give several characterizations of these domains of holomorphy (theorem of Cartan-Thullen, Levi's problem). We finally construct the holomorphic hull H(G} for each domain G, that is the largest (not necessarily schlicht) domain over en into which each function holomorphic on G can be continued.

Introduction to Complex Analysis in Several Variables

Introduction to Complex Analysis in Several Variables
Author: Volker Scheidemann
Publisher: Springer Nature
Total Pages: 239
Release: 2023
Genre: Functions of complex variables
ISBN: 3031264282

This book gives a comprehensive introduction to complex analysis in several variables. While it focusses on a number of topics in complex analysis rather than trying to cover as much material as possible, references to other parts of mathematics such as functional analysis or algebras are made to help broaden the view and the understanding of the chosen topics. A major focus are extension phenomena alien to the one-dimensional theory, which are expressed in the famous Hartog's Kugelsatz, the theorem of Cartan-Thullen, and Bochner's theorem. The book aims primarily at students starting to work in the field of complex analysis in several variables and instructors preparing a course. To that end, a lot of examples and supporting exercises are provided throughout the text. This second edition includes hints and suggestions for the solution of the provided exercises, with various degrees of support.

Functions of Several Complex Variables and Their Singularities

Functions of Several Complex Variables and Their Singularities
Author: Wolfgang Ebeling
Publisher: American Mathematical Soc.
Total Pages: 334
Release: 2007
Genre: Mathematics
ISBN: 0821833197

The book provides an introduction to the theory of functions of several complex variables and their singularities, with special emphasis on topological aspects. The topics include Riemann surfaces, holomorphic functions of several variables, classification and deformation of singularities, fundamentals of differential topology, and the topology of singularities. The aim of the book is to guide the reader from the fundamentals to more advanced topics of recent research. All the necessary prerequisites are specified and carefully explained. The general theory is illustrated by various examples and applications.

Several Complex Variables

Several Complex Variables
Author: Raghavan Narasimhan
Publisher: University of Chicago Press
Total Pages: 185
Release: 1971
Genre: Mathematics
ISBN: 0226568172

Drawn from lectures given by Raghavan Narasimhan at the University of Geneva and the University of Chicago, this book presents the part of the theory of several complex variables pertaining to unramified domains over C . Topics discussed are Hartogs' theory, domains in holomorphy, and automorphism of bounded domains.

Function Theory of Several Complex Variables

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

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.

Topics in Several Complex Variables

Topics in Several Complex Variables
Author: Zair Ibragimov
Publisher: American Mathematical Soc.
Total Pages: 168
Release: 2016-04-21
Genre: Mathematics
ISBN: 1470419270

This volume contains the proceedings of the Special Session on Several Complex Variables, which was held during the first USA-Uzbekistan Conference on Analysis and Mathematical Physics from May 20–23, 2014, at California State University, Fullerton. This volume covers a wide variety of topics in pluripotential theory, symplectic geometry and almost complex structures, integral formulas, holomorphic extension, and complex dynamics. In particular, the reader will find articles on Lagrangian submanifolds and rational convexity, multidimensional residues, S-parabolic Stein manifolds, Segre varieties, and the theory of quasianalytic functions.

Elementary Theory of Analytic Functions of One or Several Complex Variables

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

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.

Boundary Behavior of Holomorphic Functions of Several Complex Variables. (MN-11)

Boundary Behavior of Holomorphic Functions of Several Complex Variables. (MN-11)
Author: Elias M. Stein
Publisher: Princeton University Press
Total Pages: 83
Release: 2015-03-08
Genre: Mathematics
ISBN: 1400871263

This book has as its subject the boundary value theory of holomorphic functions in several complex variables, a topic that is just now coming to the forefront of mathematical analysis. For one variable, the topic is classical and rather well understood. In several variables, the necessary understanding of holomorphic functions via partial differential equations has a recent origin, and Professor Stein's book, which emphasizes the potential-theoretic aspects of the boundary value problem, should become the standard work in the field. Originally published in 1972. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Several Complex Variables IV

Several Complex Variables IV
Author: Semen G. Gindikin
Publisher: Springer Science & Business Media
Total Pages: 257
Release: 2012-12-06
Genre: Mathematics
ISBN: 3642612636

This volume of the EMS contains four survey articles on analytic spaces. They are excellent introductions to each respective area. Starting from basic principles in several complex variables each article stretches out to current trends in research. Graduate students and researchers will find a useful addition in the extensive bibliography at the end of each article.

Holomorphic Functions and Integral Representations in Several Complex Variables

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

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.