Hartogs's theorem on separate holomorphicity: Difference between revisions

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{{dablink|Note that the terminology is inconsistent and Hartogs's theorem may also mean [[Hartogs's lemma]] on removable singularities, the result on [[Hartogs number]] in axiomatic set theory, or [[Hartogs extension theorem]].}}

In [[mathematics]], '''Hartogs's theorem''' is a fundamental result of [[Friedrich Hartogs]] in the theory of [[several complex variables]]. It states that for complex-valued functions ''F'' on '''C'''^''n'', with ''n'' > 1, being an [[analytic function]] in each variable ''z''_''i'', 1 ≤ ''i'' ≤ ''n'', while the others are held constant, is enough to prove that ''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.