Theory of Stein Spaces:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer New York
1979
|
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
236 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | 1. The classical theorem of Mittag-Leffler was generalized to the case of several complex variables by Cousin in 1895. In its one variable version this says that, if one prescribes the principal parts of a merom orphic function on a domain in the complex plane e, then there exists a meromorphic function defined on that domain having exactly those principal parts. Cousin and subsequent authors could only prove the analogous theorem in several variables for certain types of domains (e. g. product domains where each factor is a domain in the complex plane). In fact it turned out that this problem can not be solved on an arbitrary domain in em, m ~ 2. The best known example for this is a "notched" bicylinder in 2 2 e . This is obtained by removing the set { (z , z ) E e 11 z I ~ !, I z 1 ~ !}, from 1 2 1 2 2 the unit bicylinder, ~ :={(z , z ) E e llz1 < 1, lz1 < 1}. This domain D has 1 2 1 2 the property that every function holomorphic on D continues to a function holo morphic on the entire bicylinder. Such a phenomenon never occurs in the theory of one complex variable. In fact, given a domain G c e, there exist functions holomorphic on G which are singular at every boundary point of G. |
| Beschreibung: | 1 Online-Ressource (XXI, 252 p) |
| ISBN: | 9781475743579 9781475743593 |
| ISSN: | 0072-7830 |
| DOI: | 10.1007/978-1-4757-4357-9 |
Internformat
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| 500 | |a 1. The classical theorem of Mittag-Leffler was generalized to the case of several complex variables by Cousin in 1895. In its one variable version this says that, if one prescribes the principal parts of a merom orphic function on a domain in the complex plane e, then there exists a meromorphic function defined on that domain having exactly those principal parts. Cousin and subsequent authors could only prove the analogous theorem in several variables for certain types of domains (e. g. product domains where each factor is a domain in the complex plane). In fact it turned out that this problem can not be solved on an arbitrary domain in em, m ~ 2. The best known example for this is a "notched" bicylinder in 2 2 e . This is obtained by removing the set { (z , z ) E e 11 z I ~ !, I z 1 ~ !}, from 1 2 1 2 2 the unit bicylinder, ~ :={(z , z ) E e llz1 < 1, lz1 < 1}. This domain D has 1 2 1 2 the property that every function holomorphic on D continues to a function holo morphic on the entire bicylinder. Such a phenomenon never occurs in the theory of one complex variable. In fact, given a domain G c e, there exist functions holomorphic on G which are singular at every boundary point of G. | ||
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Datensatz im Suchindex
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| author | Grauert, Hans |
| author_facet | Grauert, Hans |
| author_role | aut |
| author_sort | Grauert, Hans |
| author_variant | h g hg |
| building | Verbundindex |
| bvnumber | BV042421613 |
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| dewey-full | 510 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510 |
| dewey-search | 510 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-1-4757-4357-9 |
| format | Electronic eBook |
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| id | DE-604.BV042421613 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:09Z |
| institution | BVB |
| isbn | 9781475743579 9781475743593 |
| issn | 0072-7830 |
| language | English |
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| publishDate | 1979 |
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| series2 | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |
| spelling | Grauert, Hans Verfasser aut Theory of Stein Spaces by Hans Grauert, Reinhold Remmert New York, NY Springer New York 1979 1 Online-Ressource (XXI, 252 p) txt rdacontent c rdamedia cr rdacarrier Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 236 0072-7830 1. The classical theorem of Mittag-Leffler was generalized to the case of several complex variables by Cousin in 1895. In its one variable version this says that, if one prescribes the principal parts of a merom orphic function on a domain in the complex plane e, then there exists a meromorphic function defined on that domain having exactly those principal parts. Cousin and subsequent authors could only prove the analogous theorem in several variables for certain types of domains (e. g. product domains where each factor is a domain in the complex plane). In fact it turned out that this problem can not be solved on an arbitrary domain in em, m ~ 2. The best known example for this is a "notched" bicylinder in 2 2 e . This is obtained by removing the set { (z , z ) E e 11 z I ~ !, I z 1 ~ !}, from 1 2 1 2 2 the unit bicylinder, ~ :={(z , z ) E e llz1 < 1, lz1 < 1}. This domain D has 1 2 1 2 the property that every function holomorphic on D continues to a function holo morphic on the entire bicylinder. Such a phenomenon never occurs in the theory of one complex variable. In fact, given a domain G c e, there exist functions holomorphic on G which are singular at every boundary point of G. Mathematics Mathematics, general Mathematik Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Stein-Raum (DE-588)4183071-4 gnd rswk-swf Riemannsche Fläche (DE-588)4049991-1 gnd rswk-swf Komplexer Raum (DE-588)4138419-2 gnd rswk-swf Stein-Raum (DE-588)4183071-4 s 1\p DE-604 Funktionentheorie (DE-588)4018935-1 s 2\p DE-604 Riemannsche Fläche (DE-588)4049991-1 s 3\p DE-604 Komplexer Raum (DE-588)4138419-2 s 4\p DE-604 Remmert, Reinhold Sonstige oth https://doi.org/10.1007/978-1-4757-4357-9 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Grauert, Hans Theory of Stein Spaces Mathematics Mathematics, general Mathematik Funktionentheorie (DE-588)4018935-1 gnd Stein-Raum (DE-588)4183071-4 gnd Riemannsche Fläche (DE-588)4049991-1 gnd Komplexer Raum (DE-588)4138419-2 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4183071-4 (DE-588)4049991-1 (DE-588)4138419-2 |
| title | Theory of Stein Spaces |
| title_auth | Theory of Stein Spaces |
| title_exact_search | Theory of Stein Spaces |
| title_full | Theory of Stein Spaces by Hans Grauert, Reinhold Remmert |
| title_fullStr | Theory of Stein Spaces by Hans Grauert, Reinhold Remmert |
| title_full_unstemmed | Theory of Stein Spaces by Hans Grauert, Reinhold Remmert |
| title_short | Theory of Stein Spaces |
| title_sort | theory of stein spaces |
| topic | Mathematics Mathematics, general Mathematik Funktionentheorie (DE-588)4018935-1 gnd Stein-Raum (DE-588)4183071-4 gnd Riemannsche Fläche (DE-588)4049991-1 gnd Komplexer Raum (DE-588)4138419-2 gnd |
| topic_facet | Mathematics Mathematics, general Mathematik Funktionentheorie Stein-Raum Riemannsche Fläche Komplexer Raum |
| url | https://doi.org/10.1007/978-1-4757-4357-9 |
| work_keys_str_mv | AT grauerthans theoryofsteinspaces AT remmertreinhold theoryofsteinspaces |