The present book grew out of introductory lectures on the theory of functions 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.

Chapter 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

Chapter 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 Cn

 8 Holomorphic Hulls

Chapter 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

Chapter Ⅳ Sheaf Theory

 1 Sheaves of Sets

 2 Sheaves with Algebraic Structure

 3 Analytic Sheaf Morphisms

 4 Coherent Sheaves

Chapter Ⅴ Complex Manifolds

 1 Complex Ringed Spaces

 2 Function Theory on Complex Manifolds

 3 Examples of Complex Manifolds

 4 Closures of Cn

Chapter Ⅵ Cohomology Theory

 1 Flabby Cohomology

 2 The Cech Cohomology

 3 Double Complexes

 4 The Cohomology Sequence

 5 Main Theorem on Stein Manifolds

Chapter 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

Bibliography

Index