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→Formal statement: False: e.g., C\{0} cross C Tag: section blanking |
Undid revision 840672412 by 140.180.250.151 (talk) Readded correct statement. {0}xC is not an isolated singularity: it is an entire complex line |
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In fact, using the [[Cauchy integral formula]] we obtain the extended function <math>F</math> . All holomorphic functions are analytically continued to the polydisk, which is strictly larger than the ___domain on which the original holomorphic function is defined. Such phenomena never happen in the case of one variable.
==Formal statement==
:Let {{mvar|f}} be a [[holomorphic function]] on a [[Set (mathematics)|set]] {{math|''G\K''}}, where {{mvar|G}} is an open subset of {{math|'''C'''<sup>''n''</sup>}} ({{math|''n'' ≥ 2}}) and {{mvar|K}} is a compact subset of {{mvar|G}}. If the [[Complement (set theory)|complement]] {{math|''G\K''}} is connected, then {{mvar|f}} can be extended to a unique holomorphic function on {{mvar|G}}.
==Counterexamples in dimension one==
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