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Version 1 |
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\theoremstyle{definition} |
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\newtheorem*{defn}{Definition} |
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| \begin{defn} |
\begin{defn} |
| Let $a, b \in {\mathbb{C}}^n$. The set |
Let $a, b \in {\mathbb{C}}^n$. The set |
| $\ell := \{a + b z \mid z \in {\mathbb{C}} \}$ is called the {\em complex line}. |
$\ell := \{a + b z \mid z \in {\mathbb{C}} \}$ is called the {\em complex line}. |
| \end{defn} |
\end{defn} |
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| A complex line is a holomorphic complex affine imbedding of ${\mathbb{C}}$ |
A complex line is a holomorphic complex affine imbedding of ${\mathbb{C}}$ |
| into ${\mathbb{C}}^n$ so that if $f$ is holomorphic, then |
into ${\mathbb{C}}^n$ so that if $f$ is holomorphic, then |
| $z \mapsto f(a + b z)$ is also holomorphic. That is the complex structures of $\ell$ and ${\mathbb{C}}^n$ are compatible. Hence not every two dimensional real affine space is a complex line. |
$z \mapsto f(a + b z)$ is also holomorphic. That is the complex structures of $\ell$ and ${\mathbb{C}}^n$ are compatible. Hence not every two dimensional real affine space is a complex line. |
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| \begin{defn} |
\begin{defn} |
| Let $a, b_1, \ldots, b_k \in {\mathbb{C}}^n$ such that |
Let $a, b_1, \ldots, b_k \in {\mathbb{C}}^n$ such that |
| $b_1, \ldots, b_k$ are linearly independent |
$b_1, \ldots, b_k$ are linearly independent |
| over ${\mathbb{C}}$, |
over ${\mathbb{C}}$, |
| then. The set |
then. The set |
| \begin{equation*} |
\begin{equation*} |
| \{a + \sum_{j=1}^k b_k z_k \mid z_1,\ldots,z_k \in {\mathbb{C}} \} |
\{a + \sum_{j=1}^k b_k z_k \mid z_1,\ldots,z_k \in {\mathbb{C}} \} |
| \end{equation*} |
\end{equation*} |
| is called the {\em complex affine space}. |
is called the {\em complex affine space}. |
| \end{defn} |
\end{defn} |
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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} |
