If $u$ is a harmonic and $g$ is complex analytic, then why is $u(g(z))$ also harmonic.

Question

I tried to do the second derivative of it to satisfy the harmonic equation of $u_{xx} + u_{yy} = 0$. Second derivative is $u'(g(z) * g''(z) + (g''(z))^2 * u''(g(z))$.

Answer 1

A function $u: \mathbb C \to \mathbb R$ is harmonic if and only if there exist an analytic function $f$ on $\mathbb C$ such that $u(z)$ is the real part of $f(z)$ for each $z$.

Let $f$ be an analytic function on $ \mathbb C$ such that $u =\Re f$. Then $f(g(z))$ is analytic and its real part is exactly $u(g(z))$. Hence $u(gz))$ is harmonic.