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{{short description|Mathematical theorem}}
{{about|a theorem on separate holomorphicity|a theorem on extensions of holomorphic functions|Hartogs's extension theorem|a theorem on infinite ordinals|Hartogs number||}}
{{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 [[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''<sub>''i''</sub>, 1 &le; ''i'' &le; ''n'', while the other variables are held constant, then ''F'' is a [[continuous function]].
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''<sub>''i''</sub>, 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.
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.


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]].
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
There is no analogue of this [[theorem]] for [[Function of several real variables|real variables]]. If we assume that a function
<math>f \colon {\textbf{R}}^n \to {\textbf{R}}</math>
<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
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 ==
== References ==
* Steven G. Krantz. ''Function Theory of Several Complex Variables'', AMS Chelsea Publishing, Providence, Rhode Island, 1992.
* [[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==
==External links==
* {{springer|title=Hartogs theorem|id=p/h046650}}
* {{Eom|title=Hartogs theorem|oldid=40754}}


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


[[Category:Several complex variables]]
[[Category:Several complex variables]]

Latest revision as of 07:48, 30 July 2024

In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if is a function which is analytic in each variable zi, 1 ≤ in, 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.

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 real variables. If we assume that a function is differentiable (or even analytic) in each variable separately, it is not true that will necessarily be continuous. A counterexample in two dimensions is given by

If in addition we define , this function has well-defined partial derivatives in and at the origin, but it is not continuous at origin. (Indeed, the limits along the lines and are not equal, so there is no way to extend the definition of to include the origin and have the function be continuous there.)

References

[edit]
  • Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
  • Fuks, Boris Abramovich (1963). Theory of Analytic Functions of Several Complex Variables. American Mathematical Society. ISBN 978-1-4704-4428-0.
  • Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables (3rd ed.), North Holland, ISBN 978-1-493-30273-4
[edit]

This article incorporates material from Hartogs's theorem on separate analyticity on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.