Introduction to Complex Analysis in Several Variables

✍ Scribed by Volker Scheidemann

• Publisher: Birkhäuser
• Year: 2023
• Tongue: English
• Leaves: 239
• Series: Compact Textbooks in Mathematics
• Edition: 2
• Category: Library

✦ Synopsis

This book gives a comprehensive introduction to complex analysis in several variables. While it focusses on a number of topics in complex analysis rather than trying to cover as much material as possible, references to other parts of mathematics such as functional analysis or algebras are made to help broaden the view and the understanding of the chosen topics. A major focus are extension phenomena alien to the one-dimensional theory, which are expressed in the famous Hartog's Kugelsatz, the theorem of Cartan-Thullen, and Bochner's theorem.

The book aims primarily at students starting to work in the field of complex analysis in several variables and instructors preparing a course. To that end, a lot of examples and supporting exercises are provided throughout the text.
This second edition includes hints and suggestions for the solution of the provided exercises, with various degrees of support.

✦ Table of Contents

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

✦ Subjects

Higher-dimensional Complex Analysis