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Riemann's removable singularity theorem in several variables
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(Theorem)
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Theorem 1 Suppose $V$ is a proper analytic variety in an open set $U \subset {\mathbb{C}}^n$ (that is of dimension at most $n-1$ ) suppose that $f \colon U \backslash V \to {\mathbb{C}}$ is holomorphic and further that $f$ is locally bounded in $U$ Then there exists a unique holomorphic
extention of $f$ to all of $U$ .
If $V$ is of even lower dimension we can in fact even drop the locally bounded requirement, see the Hartogs extension theorem.
- 1
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
- 2
- Hassler Whitney. Complex Analytic Varieties. Addison-Wesley, Philippines, 1972.
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"Riemann's removable singularity theorem in several variables" is owned by jirka.
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| Other names: |
Riemann's extension theorem |
This object's parent.
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Cross-references: Hartogs extension theorem, even, locally bounded, holomorphic, dimension, open set, analytic variety
There is 1 reference to this entry.
This is version 1 of Riemann's removable singularity theorem in several variables, born on 2005-11-17.
Object id is 7492, canonical name is RiemannsRemovableSingularityTheoremInSeveralVariables.
Accessed 2671 times total.
Classification:
| AMS MSC: | 30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory) | | | 32H02 (Several complex variables and analytic spaces :: Holomorphic mappings and correspondences :: Holomorphic mappings, embeddings and related questions) |
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Pending Errata and Addenda
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