Hartogs's theorem on separate holomorphicity: Difference between revisions

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

In [[mathematics]], '''Hartogs' theorem''' is a fundamental result of [[Friedrich Hartogs]] in the theory of [[several complex variables]]. ItRoughly speaking, it states that fora complex-valued functions'separately ''F'analytic' onfunction is continuous. More precisely, if '''C'''<supmath>''F:{\textbf{C}}^n'' \to {\textbf{C}}</supmath>, with ''n'' > 1, beingis an [[analytic function]] in each variable ''z''_''i'', 1 &le; ''i'' &le; ''n'', while the othersother variables are held constant, is enough to prove thatthen ''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.

Note that there is no analogue of this [[theorem]] for [[real number|real]] variables. If we assume that a function

:<math>f \colon {\mathbbtextbf{R}}^n \to {\mathbbtextbf{R}}</math>

is [[differentiable]] (or even [[analytic function|analytic]]) in each variable separately, it is not true that <math>f</math> will necessarily be continuous. A counterexample in two dimensions is given by