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1 Elementary theory of several complex variables 1

1.1 Geometry of Cn 1

1.2 Holomorphic functions in several complex variables 7

1.2.1 Definition of a holomorphic function 7

1.2.2 Basic properties of holomorphic functions 10

1.2.3 Partially holomorphic functions and the Canchy-Riemann differential equations 13

1.3 The Cauchy Integral Formula 17

1.4 O(U)as a topological space 19

1.4.1 Locally convex spaces 20

1.4.2 The compact-open topology on C(U,E) 23

1.4.3 The Theorems of Arzelà-Ascoli and Montel 28

1.5 Power series and Taylor series 34

1.5.1 Summable families in Banach spaces 34

1.5.2 Power series 35

1.5.3 Reinhardt domains and Laurent expansion 38

2 Continuation on circular and polycircular domains 47

2.1 Holomorphic continuation 47

2.2 Representation-theoretic interpretation of the Laurent series 54

2.3 Hartogs' Kugelsatz,Special case 56

3 Biholomorphic maps 59

3.1 The Inverse Function Theorem and Implicit Functions 59

3.2 The Riemann Mapping Problem 64

3.3 Cartan's Uniqueness Theorem 67

4 Analytic Sets 71

4.1 Elementary properties of analytic sets 71

4.2 The Riemann Removable Singularity Theorems 75

5 Hartogs' Kugelsatz 79

5.1 Holomorphic Differential Forms 79

5.1.1 Multilinear forms 79

5.1.2 Complex differential forms 82

5.2 The inhomogenous Cauchy-Riemann Differential Equations 88

5.3 Dolbeaut's Lemma 90

5.4 The Kugelsatz of Hartogs 94

6 Continuation on Tubular Domain 97

6.1 Convex hulls 97

6.2 Holomorphically convex hulls 100

6.3 Bochner's Theorem 106

7 Cartan-Thullen Theory 111

7.1 Holomorphically convex sets 111

7.2 Domains of Holomorphy 115

7.3 The Theorem of Cartan-Thullen 118

7.4 Holomorphically convex Reinhardt domains 121

8 Local Properties of holomorphic functions 125

8.1 Local representation of a holomorphic function 125

8.1.1 Germ of a holomorphic function 125

8.1.2 The algebras of formal and of convergent power series 127

8.2 The Weierstrass Theorems 135

8.2.1 The Weierstrass Division Formula 138

8.2.2 The Weierstrass Preparation Theorem 142

8.3 Algebraic properties of C{z1,...,zn} 145

8.4 Hilbert's Nullstellensatz 151

8.4.1 Germs of a set 152

8.4.2 The radical of an ideal 156

8.4.3 Hilbert's Nullstellensatz for principal ideals 160

Register of Symbols 165

Bibliography 167

Index 169