Show that $xu_x+yu_y$ is the real part to an analytic function if $u(x,y)$ is the real part to an analytic function. To what analytic function is the real part to if $u = Re(f(z))$?
My attempted at a solution:
Write $u_1 = x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}$ and evaluate if $$\frac{\partial^2 u_1}{\partial x^2} + \frac{\partial^2 u_1}{\partial y^2}$$
is 0. However, I get third order derivatives that I can't handle. Is there a not-so-dirty way to solve this?
Hint: If $f=u+iv$ is analytic, then $f' = u_x + iv_x.$ Thefore $f'= u_x-iu_y$ by the Cauchy-Riemann equations.