Introduction to Complex Analysis in Several Variables

Preface to the Second Edition
Preface to the First Edition
Contents
Register of Symbols
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 Cauchy–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
			The Exterior Derivative
	5.2 The inhomogenous Cauchy–Riemann Differential Equations
	5.3 Dolbeault\'s Lemma
	5.4 The Kugelsatz of Hartogs
6 Continuation on Tubular Domains
	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
9 Hints to the Exercises
	9.1 Exercises in Chap.1
	9.2 Exercises in Chap.2
	9.3 Exercises in Chap.3
	9.4 Exercises in Chap.4
	9.5 Exercises in Chap.5
	9.6 Exercises in Chap.6
	9.7 Exercises in Chap.7
	9.8 Exercises in Chap.8
References