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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 $F:{\textbf {C}}^{n}\to {\textbf {C}}$ is a function which is analytic in each variable z_i, 1 ≤ i ≤ 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.

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 $f\colon {\textbf {R}}^{n}\to {\textbf {R}}$ is differentiable (or even analytic) in each variable separately, it is not true that $f$ will necessarily be continuous. A counterexample in two dimensions is given by

f(x,y)={\frac {xy}{x^{2}+y^{2}}}.

If in addition we define $f(0,0)=0$ , this function has well-defined partial derivatives in $x$ and $y$ at the origin, but it is not continuous at origin. (Indeed, the limits along the lines $x=y$ and $x=-y$ are not equal, so there is no way to extend the definition of $f$ 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.
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

External links

"Hartogs theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

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

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+{{short description|Mathematical theorem}}
-{{dablink|Note that the terminology is inconsistent and Hartogs's theorem may also mean [[Hartogs's 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 [[several complex variables]]. It states that for complex-valued functions ''F'' on '''C'''<sup>''n''</sup>, with ''n'' > 1, being an [[analytic function]] in each variable ''z''<sub>''i''</sub>, 1 &le; ''i'' &le; ''n'', while the others are held constant,  is enough to prove that ''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]] 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.
+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]].
-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>
+There is no analogue of this [[theorem]] for [[Function of several real variables|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 {\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>
-This function has well-defined [[partial derivative]]s in <math>x</math> and <math>y</math> at 0, but it is not [[Continuous function|continuous]] at 0 (the [[limit of a function|limits]] along the lines <math>x=y</math> and <math>x=-y</math> give different results).
+If 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>  are 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.
+* [[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|id=6024|title=Hartogs's theorem on separate analyticity}}
+* {{Eom|title=Hartogs theorem|oldid=40754}}
+{{PlanetMath attribution|urlname=HartogssTheoremOnSeparateAnalyticity|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]]