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Let $g$ be a holomorphic function on a domain. Show that both the real and imaginary parts of $f(z):=g(z^*)$ are harmonic.

Knowing that $g$ is holomorphic, I know that both the CR equations hold and that Laplace's condition hold for both real and imaginary parts, but can't get to the second part.

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Let $g(z)=u(x,y)+iv(x,y)$. Then $$ g(z^*)=u(x,-y)+iv(x,-y) $$ Since $g(z)$ is analytic, by Cauchy-Riemann equation, we have $$ u_x(x,-y)=-v_y(x,-y)\quad\text{and}\quad v_x(x,-y)=u_y(x,-y) $$ So $$ u_{xx}=-\frac{\partial v_y(x,-y)}{\partial x}=-v_{yx}\quad\text{and}\quad u_{yy}=\frac{\partial v_x(x,-y)}{\partial y}=v_{xy}=v_{yx} $$ i.e. $u_{xx}+u_{yy}=0$, and thus $u$ is harmonic. Likewise we can prove that $v$ is harmonic.

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