Proove that $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ is the real part of an analytic function.

Question

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?

Answer 1

Like S. Panja said, you do not have to find a harmonic conjugate, just verify the C-R equations using the C-R equations for $f=u+iv$ in the form of the Laplace equation for $u$: $u_{xx}+u_{yy}=0$.

If you want an explicit conjugate, think about what form it should have: it must somehow be "like" $U$ but with the $x$ and $y$ multipliers mixed up (to mimic the situation with the C-R equations).

If you are still having trouble, try:

$V = yu_x-xu_y$