Hartogs' theorem

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In mathematics, Hartogs' 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 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 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 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.

External links[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.[dead link]. But url=http://planetmath.org/hartogsstheoremonseparateanalyticity is available.