Satz von Hartogs über getrennte Holomorphizität

Satz von Hartogs über getrennte Holomorphizität (Redirected from Hartogs's theorem) Zur Navigation springen Zur Suche springen "Hartogs's theorem" leitet hier weiter. For the theorem on extensions of holomorphic functions, see Hartogs's extension theorem. For the theorem on infinite ordinals, see Hartogs number.

In Mathematik, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Grob gesprochen, it states that a 'separately analytic' function is continuous. Etwas präziser, wenn {Anzeigestil F:{textbf {C}}^{n}zu {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 (d.h. that locally it has a Taylor expansion). Deswegen, '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}zu {textbf {R}}} is differentiable (or even analytic) in each variable separately, it is not true that {Anzeigestil f} will necessarily be continuous. A counterexample in two dimensions is given by {Anzeigestil f(x,j)={frac {xy}{x^{2}+y^{2}}}.} If in addition we define {Anzeigestil f(0,0)=0} , this function has well-defined partial derivatives in {Anzeigestil x} und {Anzeigestil y} at the origin, but it is not continuous at origin. (In der Tat, the limits along the lines {Anzeigestil x=y} und {displaystyle x=-y} are not equal, so there is no way to extend the definition of {Anzeigestil f} to include the origin and have the function be continuous there.) References Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Verlag, Vorsehung, Rhode Island, 1992. Fuks, Boris Abramovich (1963). Theory of Analytic Functions of Several Complex Variables. ISBN 978-1-4704-4428-0. Externe Links "Hartogs theorem", Enzyklopädie der Mathematik, EMS-Presse, 2001 [1994] This article incorporates material from Hartogs's theorem on separate analyticity on PlanetMath, das unter der Creative Commons Attribution/Share-Alike License lizenziert ist.

Kategorien: Several complex variablesTheorems in complex analysis