Riemann’s removable singularity theorem in several variables

Theorem.

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 holomorphicextention of $f$ to all of $U$ .

If $V$ is of even lower dimension we can in fact even drop the locally boundedrequirement, see the Hartogs extension theorem.

References

1 Steven G. Krantz.,AMS Chelsea Publishing, Providence, Rhode Island, 1992.
2 Hassler Whitney..Addison-Wesley, Philippines, 1972.

单词	RiemannsRemovableSingularityTheoremInSeveralVariables
释义	Riemann’s removable singularity theorem in several variables Theorem. 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 holomorphicextention of $f$ to all of $U$ . If $V$ is of even lower dimension we can in fact even drop the locally boundedrequirement, see the Hartogs extension theorem. References 1 Steven G. Krantz.,AMS Chelsea Publishing, Providence, Rhode Island, 1992. 2 Hassler Whitney..Addison-Wesley, Philippines, 1972.
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