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Suppose $f(z) = u + i v$ and $F(z) = U + i V$ are entire. Show that $u(U(x,y), V(x,y)$ is harmonic everywhere.

I know that the two conditions of a harmonic equation are that all the second partial derivatives exist and that the equation satisfies the Laplace Equation $$u_{xx} + u_{yy} = 0,$$ but I don't know how to take the partial derivatives in such a manner in order to prove this.

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Rasl parts of analytic functions are harmonic and compositions of entire functions are entire. Hence, $u( U(x,y),V(x,y))$ which is the real part of $f\circ F$ is harmonic.

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