We have the given statement If f(z) is analytic then u(x,y)=Re(f(x+iy)) is harmonic.
In order to prove/disprove it, I use the Cauchy-Riemann equations, since they define analycity:
$u_x=v_y$
$u_y=-v_x$
and the Laplace equation, to prove harmonicity: $u_{xx}+v_{yy}=0$
So,
Laplace equation on $f$ gives $Re[f(z)]_{xx}+Re[fz)]_{yy}=0 $, which is feasible, since $u_{xx}=0=u_{yy}=0$ but then we get a contradiction on the Cauchy-Riemann equations. We set $u=Re[f(x+iy)]=$ and assume that $iv=Im[f(x+iy)]$
$(u)_x=(v)_y$
$(u)_y=-(v)_x $
Then it seems correct?
Any suggestions appreciated
With Thanks