extension of analytic functions of several variables Suppose that we have two functions of several complex variables that are holomorphic on the whole euclidean space. If these two functions are equal on an open and connected subset of the REAL euclidean space, can we say that they are equal everywhere?
This is true for holomorphic functions of one complex variable, but does it hold for several variables holomorphic functions?
 A: Yes, it does hold. You can argue componentwise. I assume, without loss of generality, that the second function is $\equiv 0$, and the open subset of $\mathbb{R}^n$ on which $f$ vanishes is a neighbourhood of $0$.
For $x_2,\,\ldots,\, x_n \in \mathbb{R}$ close enough to $0$, consider
$$ h(z) = f(z,\,x_2,\,\ldots,\, x_n).$$
By the identity theorem (in one variable), $h \equiv 0$. So you know $f(z_1,\, z_2,\, \ldots,\, z_n)$ is $0$ whenever the last $n-1$ arguments are real and of small enough norm.
Then for arbitrary $z_1 \in \mathbb{C}$ and $x_3,\,\ldots,\,x_n \in \mathbb{R}$ of small enough norm, consider
$$k(w) = f(z_1,\,w,\,x_3,\,\ldots,\,x_n).$$
By the identity theorem, $k \equiv 0$.
Thus $f(z_1,\, z_2,\, \ldots,\, z_n) = 0$ whenever the last $n-2$ arguments are real and of small enough norm.
Repeat until the last argument.
A: We may assume that the second function is identically zero.
The first  function is entire on $\mathbb C^n$, and thus has a series expansion of the form $$f(z)=\sum c_Iz^I=\sum a_Iz^I+i\sum b_Iz^I$$ (where $I\in \mathbb N^n$ is a multi-index, $c_I=a_I+ib_I$ and $a_I\in \mathbb R, b_I\in \mathbb R$).    
Assuming that the open subset of $\mathbb R^n$ contains zero, we have $f(r)=\sum c_I r^I=\sum a_I r^I+i\sum b_I r^I=0$ for all sufficiently small $r=(r_1,\cdots,r_n)\in \mathbb R^n$.
This implies  $\sum a_I r^I=0,\sum b_I r^I=0$ which forces all $a_I,b_I$ to be zero.
Thus all $c_I=a_I+ib_I=0$ and $f=0$.
