TITLE: product of harmonic functions
QUESTION [5 upvotes]: "Let $u, v$ be harmonic functions on a region G. Prove that if the product $uv$ is identically zero, then either $u$ or $v$ must be identically zero." Could sb. give me a hint for this question? I tried all the properties and theorems on harmonic functions (definition, poisson integral, max. principle, passed a holomorphic function,..), but could go nowhere.
Thanks.

REPLY [4 votes]: Suppose $u$ is not identically zero. Then $u(z_1)≠0$ for some $z_1$. But this forces $v(z_1)=0$ since their product is zero. As $u$ being cts., there exists a ball around $z_1$ s.t. $u≠0$. So $v$ is zero on this ball. Hence, by the identity theorem for harmonic functions, $v$ is identically zero on $G$.