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书名 多元复分析导论
分类 科学技术-自然科学-数学
作者 (德)谢德曼
出版社 世界图书出版公司
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这本《多元复分析导论》由Volker Scheidemann著,全面介绍了多变量复分析,旨在全面清晰地介绍多变复分析,而非包揽尽可能多的相关材料。书中包括了不少相关的数学概念,如泛函分析和代数,以丰富书的内容。书中还集中讲述了一维的情况,例如著名的Hartog的Kugelsatz,Carton-Thullen定理和Bochner定理。

目录

Preface

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 Arzela-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 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

Register of Symbols

Bibliography

Index

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