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A real hyperplane of $\mathbb{C}^{n}$ is given by a equation $\textrm{Re}( \ell(Z)) = c$, where $c \in \mathbb{R}$ and $\ell(Z) = \sum_{j=1}^{n} a_{j}z_{j}, a_{j} \in \mathbb{C}$. How to prove that if $f$ is a holomorphic function of $\mathbb{C}^{n}$ that vanishes in a real hyperplane, then $f$ is identically zero?

Thank you!

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    You might consider the following. Let $L$ be a one (complex) dimensional subspace of $\mathbb{C}^n$. You know that $f|_L$ is holomorphic on $L\cong \mathbb{C}$. What does the zero set of $f$ look like in $L$? It certainly contains the intersection of $L$ and Re$(\ell(Z)) = c$. How big is this set?2012-10-16

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