Introduction to complex analysis in several variables:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel ; Boston, Mass. ; Berlin
Birkhäuser Verlag
2005
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Auch als Internetausgabe |
| Physical Description: | VIII, 171 Seiten |
| ISBN: | 376437490X 3764374918 9783764374907 |
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MARC
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| 100 | 1 | |a Scheidemann, Volker |d 1968- |e Verfasser |0 (DE-588)122279794 |4 aut | |
| 245 | 1 | 0 | |a Introduction to complex analysis in several variables |c Volker Scheidemann |
| 264 | 1 | |a Basel ; Boston, Mass. ; Berlin |b Birkhäuser Verlag |c 2005 | |
| 300 | |a VIII, 171 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Auch als Internetausgabe | ||
| 650 | 7 | |a Complexe variabelen |2 gtt | |
| 650 | 4 | |a Fonctions de plusieurs variables complexes | |
| 650 | 4 | |a Functions of several complex variables | |
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Record in the Search Index
| _version_ | 1804134590301863936 |
|---|---|
| adam_text | Contents
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
Ό
(U)
as
Ά
topologica!
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
Arzelà-Ascoli
and
Montei
......... 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
Cartau
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
......................
Ill
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
С
{z
,...
,zn}
................. 145
8.4
Hubert s Nullstellensatz
........................ 151
8.4.1
Germs of a set
......................... 152
8.4.2,
The radical of an ideal
..................... 156
8.4.3
Hubert s Nullstellensatz for principal ideals
......... 160
Register of Symbols
165
Bibliography
167
Index
169
|
| adam_txt |
Contents
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
Ό
(U)
as
Ά
topologica!
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
Arzelà-Ascoli
and
Montei
. 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
Cartau
'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
.
Ill
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
С
{z\
,.
,zn}
. 145
8.4
Hubert's Nullstellensatz
. 151
8.4.1
Germs of a set
. 152
8.4.2,
The radical of an ideal
. 156
8.4.3
Hubert's Nullstellensatz for principal ideals
. 160
Register of Symbols
165
Bibliography
167
Index
169 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T13:24:37Z |
| indexdate | 2024-07-09T20:27:02Z |
| institution | BVB |
| isbn | 376437490X 3764374918 9783764374907 |
| language | English |
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| physical | VIII, 171 Seiten |
| publishDate | 2005 |
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| publisher | Birkhäuser Verlag |
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| spelling | Scheidemann, Volker 1968- Verfasser (DE-588)122279794 aut Introduction to complex analysis in several variables Volker Scheidemann Basel ; Boston, Mass. ; Berlin Birkhäuser Verlag 2005 VIII, 171 Seiten txt rdacontent n rdamedia nc rdacarrier Auch als Internetausgabe Complexe variabelen gtt Fonctions de plusieurs variables complexes Functions of several complex variables Mehrere Variable (DE-588)4277015-4 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 s Mehrere Variable (DE-588)4277015-4 s DE-604 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014188535&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Scheidemann, Volker 1968- Introduction to complex analysis in several variables Complexe variabelen gtt Fonctions de plusieurs variables complexes Functions of several complex variables Mehrere Variable (DE-588)4277015-4 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4277015-4 (DE-588)4018935-1 |
| title | Introduction to complex analysis in several variables |
| title_auth | Introduction to complex analysis in several variables |
| title_exact_search | Introduction to complex analysis in several variables |
| title_exact_search_txtP | Introduction to complex analysis in several variables |
| title_full | Introduction to complex analysis in several variables Volker Scheidemann |
| title_fullStr | Introduction to complex analysis in several variables Volker Scheidemann |
| title_full_unstemmed | Introduction to complex analysis in several variables Volker Scheidemann |
| title_short | Introduction to complex analysis in several variables |
| title_sort | introduction to complex analysis in several variables |
| topic | Complexe variabelen gtt Fonctions de plusieurs variables complexes Functions of several complex variables Mehrere Variable (DE-588)4277015-4 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Complexe variabelen Fonctions de plusieurs variables complexes Functions of several complex variables Mehrere Variable Funktionentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014188535&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT scheidemannvolker introductiontocomplexanalysisinseveralvariables |