The problem is as follows:
Proove that $U(x,y) = x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ is the real part of an analytic function.
where $f(z)$ is analytic such that $u(x,y) = \mathrm{Re}[f(z)]$?
I've been playing around with the Cauchy-Riemann equations trying to find a harmonic conjugate, but I feel pretty stuck. Does anybody have a clue where to begin?