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]]. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if <math>F:{\textbf{C}}^n \to {\textbf{C}}</math> is ana 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]] 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.

The theoremStarting with the extra conditionhypothesis 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 thereThere 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

:<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 the origin, but it is not [[Continuous function|continuous]] at origin. (Indeed, the [[limit of a function|limits]] along the lines <math>x=y</math> and <math>x=-y</math> give differentare resultsnot andequal, fso there is notno definedway atto 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==

{{PlanetMath attribution|id=6024|title=Hartogs's theorem on separate analyticity}}{{dead link|date=September 2014}}. But url=http://planetmath.org/hartogsstheoremonseparateanalyticity is available.

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

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

[[Category:Several complex variables]]