I tried to do the second derivative of it to satisfy the harmonic equation of $u_{xx} + u_{yy} = 0$. Second derivative is $u'(g(z) * g''(z) + (g''(z))^2 * u''(g(z))$.
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$u$ is the real part of an analytic function $f$ and $u(g(z))$ is the real part of the analytic function $f(g(z))$. – Kavi Rama Murthy Oct 19 '20 at 08:28
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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.
Kavi Rama Murthy
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