# Hartogs's theorem on separate holomorphicity

Hartogs's theorem on separate holomorphicity (Redirected from Hartogs's theorem) Ir para a navegação Ir para a pesquisa "Hartogs's theorem" redireciona aqui. For the theorem on extensions of holomorphic functions, see Hartogs's extension theorem. For the theorem on infinite ordinals, see Hartogs number.

Na matemática, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. A grosso modo, it states that a 'separately analytic' function is continuous. Mais precisamente, E se {estilo de exibição F:{textbf {C}}^{n}para {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 (ou seja. that locally it has a Taylor expansion). Portanto, '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}para {textbf {R}}} is differentiable (or even analytic) in each variable separately, it is not true that {estilo de exibição f} will necessarily be continuous. A counterexample in two dimensions is given by {estilo de exibição f(x,y)={fratura {xy}{x^{2}+^{2}}}.} If in addition we define {estilo de exibição f(0,0)=0} , this function has well-defined partial derivatives in {estilo de exibição x} e {estilo de exibição y} at the origin, but it is not continuous at origin. (De fato, the limits along the lines {estilo de exibição x=y} e {displaystyle x=-y} are not equal, so there is no way to extend the definition of {estilo de exibição f} to include the origin and have the function be continuous there.) References Steven G. Krantz. Function Theory of Several Complex Variables, Editora AMS Chelsea, Providência, Rhode Island, 1992. Fuks, Boris Abramovich (1963). Theory of Analytic Functions of Several Complex Variables. ISBN 978-1-4704-4428-0. links externos "Hartogs theorem", Enciclopédia de Matemática, Imprensa EMS, 2001 [1994] This article incorporates material from Hartogs's theorem on separate analyticity on PlanetMath, que está licenciado sob a Licença Creative Commons Atribuição/Compartilhamento.

Categorias: Several complex variablesTheorems in complex analysis

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