Hartogs's theorem on separate holomorphicity: Difference between revisions

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{{redirect|Hartogs's theorem|the theorem on extensions of holomorphic functions|Hartogs's extension theorem|the theorem on infinite ordinals|Hartogs number||}}

In [[mathematics]], '''Hartogs's theorem''' is a fundamental result of [[Friedrich Hartogs]] in the theory of [[Function of several complex variables|several complex variables]]. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if <math>F:{\textbf{C}}^n \to {\textbf{C}}</math> is a function which is [[analytic function|analytic]] in each variable ''z''_''i'', 1 &le; ''i'' &le; ''n'', while the other variables are held constant, then ''F'' is a [[continuous function]].

A [[corollary]] is that the function ''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 theory of several complex variables.

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Starting with the extra hypothesis that the function is continuous (or bounded), the theorem is much easier to prove and in this form is known as [[Osgood's lemma]].

There is no analogue of this [[theorem]] for [[Function of several real numbervariables|real]] variables]]. If we assume that a function

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

is [[Differentiable function|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

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== References ==

* [[Steven G. Krantz]]. ''Function Theory of Several Complex Variables'', AMS Chelsea Publishing, Providence, Rhode Island, 1992.

* {{cite book |isbn=978-1-4704-4428-0|title=Theory of Analytic Functions of Several Complex Variables|last1=Fuks|first1=Boris Abramovich|year=1963|publisher=American Mathematical Society |url={{Google books|title=Analytic Functions of Several Complex Variables|OSlWYzf2FcwC|page=21|plainurl=yes}}}}

*{{Citation |last= Hörmander |first =Lars |authorlink =Lars Hörmander |date=1990 |orig-year=1966 |title=An Introduction to Complex Analysis in Several Variables |edition=3rd |publisher=North Holland |isbn=978-1-493-30273-4 |url={{Google books|MaM7AAAAQBAJ|An Introduction to Complex Analysis in Several Variables|plainurl=yes}}}}

==External links==

* {{springerEom|title=Hartogs theorem|idoldid=p/h04665040754}}

* http://planetmath.org/hartogsstheoremonseparateanalyticity

{{PlanetMath attribution|idurlname=6024HartogssTheoremOnSeparateAnalyticity|title=Hartogs's theorem on separate analyticity}}

[[Category:Several complex variables]]