You are here

Introduction to Complex Analysis in Several Variables

Volker Scheidemann
Publisher: 
Birkhäuser
Publication Date: 
2005
Number of Pages: 
171
Format: 
Paperback
Price: 
34.95
ISBN: 
3-7643-7490-X
Category: 
Textbook
[Reviewed by
Michaela Poplicher
, on
05/25/2006
]

This is a book written with the student in mind: it offers a concise treatment of complex analysis in several variables, by focusing on special topics rather than trying to be a comprehensive treatment of the subject. This way both the size of the book and its price are reduced, making it more affordable and attractive to students.

The book can also be considered as a possible text for a course because it contains, along with the detailed proofs of the theorems, many examples and exercises. The prerequisites are minimal for the subject: a course in complex analysis of one variable, multidimensional calculus (real variables), and an abstract algebra course are all that is required. Probably some of the exercises need some more background.

The main importance of this book is the fact that it fills a void: there are relatively few books on several complex variables. Any (graduate) student beginning work in this area and any instructor considering teaching a course in several complex variables should definitely take a look at Professor Scheidemann's work.


Mihaela Poplicher is an assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.

Preface vii

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 Cauchy–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 Arzel`a–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

vi Contents

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 Domains 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

Dummy View - NOT TO BE DELETED