- Hartogs' theorem
"NB that the terminology is inconsistent and Hartogs' theorem may also mean
Hartogs' lemma on removable singularities, or the result onHartogs number "In
mathematics , Hartogs' theorem is a fundamental result ofFriedrich Hartogs in the theory ofseveral complex variables . It states that for complex-valued functions "F" on C"n", with "n" > 1, being ananalytic function in each variable "z""i", 1 ≤ "i" ≤ "n", while the others are held constant, is enough to prove "F" acontinuous 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 aTaylor expansion ). Therefore 'separate analyticity' and 'analyticity' are coincident notions, in the several complex variables theory.Note that there is no analogue of this
theorem for real variables. If we assume that a function :f colon {mathbb{R^n o {mathbb{R isdifferentiable (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}.
This function has well-defined
partial derivative s in x and y at 0, but it is not continuous at 0 (the limits along the lines x=y and x=-y give different results).References
* Steven G. Krantz. "Function Theory of Several Complex Variables", AMS Chelsea Publishing, Providence, Rhode Island, 1992.----
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