Description
Book SynopsisIt 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.
Table of ContentsI 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.
