## holomorphic extension of functions [closed]

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hallo,

I have a problem that I dont understand. Let $U \subset \mathbb{C}$ be a open neigbourhood of zero. Let furthermore $f,g : U \subset \mathbb{C} = \mathbb{R}^{2} \rightarrow \mathbb{R}$ be two analytic functions defined by $f(x,y) = x + y$ and $g(x,y) = x$. Since these functions are analytic we can extend them holomorphically to some open neighbourhood $W$ of zero in $\mathbb{C}^{2}$ just by $f(z_{1}, z_{2}) = z_{1} + z_{2}$ and $g(z_{1}, z_{2}) = z_{1}$. Its obvious that these functions are holomorphic on $W$ since they are polynomials. Furthermore $f$ and $g$ agree on $W \cap U \cap \mathbb{R}$. From the identity Theorem they have to be the same, since they are holomorphic. But obviously they are NOT! I am wondering where the mistake is? does anybody know. I would be very thankfull for some answers.

Greetings bruno

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1
Try stating the Identity Principle for holomorphic functions in two variables... – Yemon Choi Sep 8 at 7:09
Several complex variables is a very different world from single variable. Check out Krantz's or Hormander's book. – Steven Gubkin Sep 24 at 13:12
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