# Harmonic function and harmonic conjugate

Let $$u:Gsubsetmathbb{R} rightarrow mathbb{R}$$ a harmonic function $$v:Grightarrow mathbb{R}$$ the harmonic conjugate function, with $$G$$ a domain. Prove that $$u^2-v^2$$ and $$uv$$ are harmonic without derivatives.

Before this, I proved that $$u^2$$ is harmonic, if $$u$$ is an harmonic function. Then, I thought that $$u^2$$ and $$v^2$$ are harmonic functions, and I wanted to conclude that $$u^2-v^2$$ is a harmonic function.

Nonetheless, this interpretation is wrong.