Teorema di Hartogs sull'olomorficità separata

Teorema di Hartogs sull'olomorficità separata (Redirected from Hartogs's theorem) Vai alla navigazione Vai alla ricerca "Hartogs's theorem" reindirizza qui. For the theorem on extensions of holomorphic functions, see Hartogs's extension theorem. For the theorem on infinite ordinals, see Hartogs number.

In matematica, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. In parole povere, it states that a 'separately analytic' function is continuous. Più precisamente, Se {stile di visualizzazione F:{textbf {C}}^{n}a {textbf {C}}} is a function which is analytic in each variable zi, 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 (cioè. that locally it has a Taylor expansion). Perciò, '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 {displaystyle fcolon {textbf {R}}^{n}a {textbf {R}}} is differentiable (or even analytic) in each variable separately, it is not true that {stile di visualizzazione f} will necessarily be continuous. A counterexample in two dimensions is given by {stile di visualizzazione f(X,y)={frac {xy}{x^{2}+si^{2}}}.} If in addition we define {stile di visualizzazione f(0,0)=0} , this function has well-defined partial derivatives in {stile di visualizzazione x} e {stile di visualizzazione y} at the origin, but it is not continuous at origin. (Infatti, the limits along the lines {stile di visualizzazione x=y} e {displaystyle x=-y} are not equal, so there is no way to extend the definition of {stile di visualizzazione 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, Provvidenza, Rhode Island, 1992. Fuks, Boris Abramovich (1963). Theory of Analytic Functions of Several Complex Variables. ISBN 978-1-4704-4428-0. link esterno "Hartogs theorem", Enciclopedia della matematica, EMS Press, 2001 [1994] This article incorporates material from Hartogs's theorem on separate analyticity on PlanetMath, che è concesso in licenza in base alla licenza Creative Commons Attribution/Share-Alike.

Categorie: Several complex variablesTheorems in complex analysis