show harmonic function can be the real part of an analytic one

If we can assume that the Phi(x,y) is a real valued function then this is what you do:

Apply the Cauchy-Riemann equations and show that F is analytic. Use the chain rule:

$\frac{\partial }{\partial x}f=2\left ( \frac{\partial \phi ((z+1)/2,(z-1)/2i) }{\partial ((z+1)/2)}\left ( \frac{\partial (z+1)/2}{\partial x} \right ) \right )+$
$2\left ( \frac{\partial \phi ((z+1)/2,(z-1)/2i) }{\partial ((z-1)/2i)}\left ( \frac{\partial (z-1)/2i}{\partial x} \right ) \right )+...$

and remember that z = x + iy

You will find that F is indeed analytic.