Several Complex Variables
Published on 12. December 2011
Book
Paperback/Softback
VIII, 208 pages
978-1-4612-9876-2 (ISBN)
Description
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.
More details
Series
Edition
Softcover reprint of the original 1st ed. 1976
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Research
Illustrations
VIII, 208 p.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 13 mm
Weight
341 gr
ISBN-13
978-1-4612-9876-2 (9781461298762)
DOI
10.1007/978-1-4612-9874-8
Schweitzer Classification
Other editions
Additional editions
H. Grauert | Klaus Fritzsche
Several Complex Variables
Book
12/1976
Springer
€89.13
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Content
I Holomorphic Functions.- 1 Power Series.- 2 Complex Differentiable Functions.- 3 The Cauchy Integral.- 4 Identity Theorems.- 5 Expansion in Reinhardt Domains.- 6 Real and Complex Differentiability.- 7 Holomorphic Mappings.- II Domains of Holomorphy.- 1 The Continuity Theorem.- 2 Pseudoconvexity.- 3 Holomorphic Convexity.- 4 The Thullen Theorem.- 5 Holomorphically Convex Domains.- 6 Examples.- 7 Riemann Domains over ?n.- 8 Holomorphic Hulls.- III The Weierstrass Preparation Theorem.- 1 The Algebra of Power Series.- 2 The Weierstrass Formula.- 3 Convergent Power Series.- 4 Prime Factorization.- 5 Further Consequences (Hensel Rings, Noetherian Rings).- 6 Analytic Sets.- IV Sheaf Theory.- 1 Sheaves of Sets.- 2 Sheaves with Algebraic Structure.- 3 Analytic Sheaf Morphisms.- 4 Coherent Sheaves.- V Complex Manifolds.- 1 Complex Ringed Spaces.- 2 Function Theory on Complex Manifolds.- 3 Examples of Complex Manifolds.- 4 Closures of ?n.- VI Cohomology Theory.- 1 Flabby Cohomology.- 2 The ?ech Cohomology.- 3 Double Complexes.- 4 The Cohomology Sequence.- 5 Main Theorem on Stein Manifolds.- VII Real Methods.- 1 Tangential Vectors.- 2 Differential Forms on Complex Manifolds.- 3 Cauchy Integrals.- 4 Dolbeault's Lemma.- 5 Fine Sheaves (Theorems of Dolbeault and de Rham).- List of symbols.