# Show a function is harmonic

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.

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.