1 Elementary theory of several complex variables

1.1 Geometry of Cn

1.2 Holomorphic functions in several complex variables

1.2.1 Definition of a holomorphic function

1.2.2 Basic properties of holomorphic functions

1.2.3 Partially holomorphic functions and the Canchy-Riemann differential equations

1.3 The Cauchy Integral Formula

1.4 O(U)as a topological space

1.4.1 Locally convex spaces

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

1.4.3 The Theorems of Arzelà-Ascoli and Montel

1.5 Power series and Taylor series

1.5.1 Summable families in Banach spaces

1.5.2 Power series

1.5.3 Reinhardt domains and Laurent expansion

2 Continuation on circular and polycircular domains

2.1 Holomorphic continuation

2.2 Representation-theoretic interpretation of the Laurent series

2.3 Hartogs' Kugelsatz,Special case

3 Biholomorphic maps

3.1 The Inverse Function Theorem and Implicit Functions

3.2 The Riemann Mapping Problem

3.3 Cartan's Uniqueness Theorem

4 Analytic Sets

4.1 Elementary properties of analytic sets

4.2 The Riemann Removable Singularity Theorems

5 Hartogs' Kugelsatz

5.1 Holomorphic Differential Forms

5.1.1 Multilinear forms

5.1.2 Complex differential forms

5.2 The inhomogenous Cauchy-Riemann Differential Equations

5.3 Dolbeaut's Lemma

5.4 The Kugelsatz of Hartogs

6 Continuation on Tubular Domain

6.1 Convex hulls

6.2 Holomorphically convex hulls

6.3 Bochner's Theorem

7 Cartan-Thullen Theory

7.1 Holomorphically convex sets

7.2 Domains of Holomorphy

7.3 The Theorem of Cartan-Thullen

7.4 Holomorphically convex Reinhardt domains

8 Local Properties of holomorphic functions

8.1 Local representation of a holomorphic function

8.1.1 Germ of a holomorphic function

8.1.2 The algebras of formal and of convergent power series

8.2 The Weierstrass Theorems

8.2.1 The Weierstrass Division Formula

8.2.2 The Weierstrass Preparation Theorem

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

8.4 Hilbert's Nullstellensatz

8.4.1 Germs of a set

8.4.2 The radical of an ideal

8.4.3 Hilbert's Nullstellensatz for principal ideals

Register of Symbols

Bibliography

Index