according to this website the following theorem holds true for functions of several complex variables:
If $f(z)$ is an analytic function on a domain $D\subset C^n$ that vanishes in a real neighbourhood $U$ of a point $z_0=x_0+iy_0\in D$, that is, on a set $U=\{z=x+iy\in C^n:|x−x_0|<r,y=y_0\}$, then $f=0$ on $D$.
I couldn't find a reference for this theorem. Do you know a reference resp. an easy way to show this result?