Suppose that $v$ is a harmonic conjugate for $u$ on a domain $D$. Prove that $u(x,y)^3 - 3u(x,y)v(x,y)^2$ is harmonic.
I'm trying to prove that this function is also harmonic when $v$ is harmonic conjugate for $u$. I know that when this is the case, the function $f(z) = u(x,y) + iv(x,y)$ is analytic, and thus, $u+v$ or $uv$ are also harmonic. But I am not sure how to deal with the powers. Please help!
Thank you
Suppose $u$ is harmonic (and real valued) and $v$ is a harmonic conjugate of $v$. Then $u + iv$ is analytic. Thus $(u + iv)^3$ is analytic. Thus the real part of $(u + iv)^3$ is harmonic. Because the real part of $(u + iv)^3$ equals $u^3 - 3uv^2$, we conclude that $u^3 - 3uv^2$ is harmonic.