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:
$\displaystyle \frac{\partial }{\partial x}f=2\left ( \frac{\partial \phi ((z+1)/2,(z-1)/2i) }{\partial ((z+1)/2)}\left ( \frac{\partial (z+1)/2}{\partial x} \right ) \right )+$
$\displaystyle 2\left ( \frac{\partial \phi ((z+1)/2,(z-1)/2i) }{\partial ((z-1)/2i)}\left ( \frac{\partial (z-1)/2i}{\partial x} \right ) \right )+...$
and remember that z = x + iy
You will find that F is indeed analytic.