On R. Michael range, Holomorphic Functions and Integral Representations in several Complex Variables there is a problem saying that:
Let $D\subset \mathbb{C}^n$ be connected and suppose $f:D\times D\rightarrow \mathbb{C}$ is holomorphic in the $2n$ complex variables $(z,w)\in D\times D$. Show that if there is a point $p\in D$ with $\overline{p}\in D$, such that $f(z,\overline{z})=0$ for all $z$ in a neighborhood of $p$, then $f(z,w)=0$ for all $(z,w)\in D\times D$.
There is a hint saying that: Introduce new coordinates $u=z+w, v=z-w$. I'm unable to use the hint. Please please give me some clue how to use this hint or another way to solve the problem.