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- Hartogs'_theorem abstract "In mathematics, Hartogs' theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if is an analytic function in each variable zi, 1 ≤ i ≤ n, while the other variables are held constant, then F is a continuous function.A corollary of this is that F is then in fact an analytic function in the n-variable sense (i.e. that locally it has a Taylor expansion). Therefore 'separate analyticity' and 'analyticity' are coincident notions, in the several complex variables theory.The theorem with the extra condition that the function is continuous (or bounded) is much easier to prove and is known as Osgood's lemma.Note that there is no analogue of this theorem for real variables. If we assume that a function is differentiable (or even analytic) in each variable separately, it is not true that will necessarily be continuous. A counterexample in two dimensions is given by This function has well-defined partial derivatives in and at the origin, but it is not continuous at origin (the limits along the lines and give different results and f is not defined at the origin).".
- Hartogs'_theorem wikiPageID "2978799".
- Hartogs'_theorem wikiPageRevisionID "603124704".
- Hartogs'_theorem hasPhotoCollection Hartogs'_theorem.
- Hartogs'_theorem id "6024".
- Hartogs'_theorem title "Hartogs's theorem on separate analyticity".
- Hartogs'_theorem subject Category:Several_complex_variables.
- Hartogs'_theorem subject Category:Theorems_in_complex_analysis.
- Hartogs'_theorem type Abstraction100002137.
- Hartogs'_theorem type Communication100033020.
- Hartogs'_theorem type Message106598915.
- Hartogs'_theorem type Proposition106750804.
- Hartogs'_theorem type Statement106722453.
- Hartogs'_theorem type Theorem106752293.
- Hartogs'_theorem type TheoremsInComplexAnalysis.
- Hartogs'_theorem comment "In mathematics, Hartogs' theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if is an analytic function in each variable zi, 1 ≤ i ≤ n, while the other variables are held constant, then F is a continuous function.A corollary of this is that F is then in fact an analytic function in the n-variable sense (i.e. that locally it has a Taylor expansion).".
- Hartogs'_theorem label "Hartogs' theorem".
- Hartogs'_theorem label "Satz von Hartogs (Funktionentheorie)".
- Hartogs'_theorem label "Teorema di Hartogs".
- Hartogs'_theorem label "Теорема Хартогса".
- Hartogs'_theorem sameAs Satz_von_Hartogs_(Funktionentheorie).
- Hartogs'_theorem sameAs Teorema_di_Hartogs.
- Hartogs'_theorem sameAs 하르톡스의_정리_(복소해석학).
- Hartogs'_theorem sameAs m.08hcmn.
- Hartogs'_theorem sameAs Q1050932.
- Hartogs'_theorem sameAs Q1050932.
- Hartogs'_theorem sameAs Hartogs'_theorem.
- Hartogs'_theorem wasDerivedFrom Hartogs'_theorem?oldid=603124704.
- Hartogs'_theorem isPrimaryTopicOf Hartogs'_theorem.