Hartogs's theorem on separate holomorphicity: Difference between revisions

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{{short description|Mathematical theorem}}

{{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]].}}

{{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]]. 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>, withis ''n''a > 1,function beingwhich anis [[analytic function|analytic]] 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 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 theory.

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]].

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

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

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

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

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

:<math>f(x,y) = \frac{xy}{x^2+y^2}.</math>

ThisIf in addition we define <math>f(0,0)=0</math>, this function has well-defined [[partial derivative]]s in <math>x</math> and <math>y</math> at 0the origin, but it is not [[Continuous function|continuous]] at 0origin. (Indeed, the [[limit of a function|limits]] along the lines <math>x=y</math> and <math>x=-y</math> give differentare results).not equal, so there is no way to extend the definition of <math>f</math> to include the origin and have the function be continuous there.)

== 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==

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* {{Eom|title=Hartogs theorem|oldid=40754}}

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

[[Category:Several complex variables]]

[[Category:Mathematical theorems]]

[[Category:Several complex variables]]

[[de:Satz von Hartogs (Funktionentheorie)]]

[[Category:Theorems in complex analysis]]