Hartogs's theorem on separate holomorphicity: Difference between revisions

Revision as of 03:31, 14 April 2012

Revision as of 17:00, 21 November 2012

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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.

The theorem with the extra condition that the function is continuous (or bounded) is much easier to prove and 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

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:<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).

== References ==