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Version 1 |
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\theoremstyle{theorem} |
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\newtheorem*{thm}{Theorem} |
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| \begin{thm}[Hartogs] |
\begin{thm}[Hartogs] |
| Let $G \subset {\mathbb{C}}^n$ be an open set and write |
Let $G \subset {\mathbb{C}}^n$ be an open set and write |
| $z = (z_1,\ldots,z_n)$. Let $f \colon G \to {\mathbb{C}}$ be a function |
$z = (z_1,\ldots,z_n)$. Let $f \colon G \to {\mathbb{C}}$ be a function |
| such that for each $k = 1,\ldots,n$ and fixed $z_1,\ldots,z_{k-1},z_{k+1},\ldots,z_n$ the function |
such that for each $k = 1,\ldots,n$ and fixed $z_1,\ldots,z_{k-1},z_{k+1},\ldots,z_n$ the function |
| \begin{equation*} |
\begin{equation*} |
| w \mapsto f(z_1,\ldots,z_{k-1},w,z_{k+1},\ldots,z_n) |
w \mapsto f(z_1,\ldots,z_{k-1},w,z_{k+1},\ldots,z_n) |
| \end{equation*} |
\end{equation*} |
| ia holomorphic on the set $\{ w \in {\mathbb{C}} \mid |
ia holomorphic on the set $\{ w \in {\mathbb{C}} \mid |
| z_1,\ldots,z_{k-1},w,z_{k+1},\ldots,z_n \in G \}$. Then |
z_1,\ldots,z_{k-1},w,z_{k+1},\ldots,z_n \in G \}$. Then |
| $f$ is continuous on $G$. |
$f$ is continuous on $G$. |
| \end{thm} |
\end{thm} |
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| This is a \PMlinkescapetext{sort} of an analogue of Goursat's theorem for several complex variables. That is if we just consider that a function is holomorphic in each |
This is a \PMlinkescapetext{sort} of an analogue of Goursat's theorem for several complex variables. That is if we just consider that a function is holomorphic in each |
| variable separately, then it will turn out to be continuously differentiable. Thus we can in fact define holomorphic functions of several complex variables to be just functions holomorphic in each variable separately. |
variable separately, then it will turn out to be continuously differentiable. Thus we can in fact define holomorphic functions of several complex variables to be just functions holomorphic in each variable separately. |
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| Note that there is no analogue of this theorem for real variables. If we |
Note that there is no analogue of this theorem for real variables. If we |
| assume that a function $f \colon {\mathbb{R}}^n \to {\mathbb{R}}$ is |
assume that a function $f \colon {\mathbb{R}}^n \to {\mathbb{R}}$ is |
| analytic in each variable separately, it is not true that $f$ will neccessarily |
analytic in each variable separately, it is not true that $f$ will neccessarily |
| be continuous. |
be continuous. |
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|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{Krantz:several} |
\bibitem{Krantz:several} |
| Steven~G.\@ Krantz. |
Steven~G.\@ Krantz. |
| {\em \PMlinkescapetext{Function Theory of Several Complex Variables}}, |
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}}, |
| AMS Chelsea Publishing, Providence, Rhode Island, 1992. |
AMS Chelsea Publishing, Providence, Rhode Island, 1992. |
| \end{thebibliography} |
\end{thebibliography} |
