The following is a qual-prep question:
Describe the set of harmonic functions $h(x,y)$on $\mathbb{C}$ s.t. $(x^2-y^2)h(x,y)$ is harmonic.
I've tried using the definition of harmonic function from which after some algebraic manipulations I can see that $h$ must satisfy $y \frac{\partial h}{\partial y} = x \frac{\partial h}{\partial x}$. But I think the answer calls for a more explicit answer and I don't know how to go from here.
Since $h$ satisfies your condition : $x h_x - yh_y = 0$ we can successively take partial derivatives with respect to $x$ and $y$ to get,
$xh_{xx} + h_x - yh_{xy} = 0 \quad \text{ and } \quad xh_{yx} - h_y - yh_{yy}= 0 $.
Using the identities $h_{xx} + h_{yy} = 0$ and $h_{xy} = h_{yx}$ we get from the above two equations,
$\quad h_x - xh_{yy} - y(\frac{1}{x}h_y + \frac{y}{x} h_{yy})=0; \quad (x\not=0)$
$\implies -(x^2+ y^2) h_{yy} = 0 \implies h_{yy} = 0 $. Similarly we can arrive at $h_{xx} = 0$.
From $h_{yy} = 0$ you can write that $h(x,y) = A(x)y + B(x)$ and then compare with $h_{xx} = 0 $ to arrive at the conclusion that $B = constant $, and $A(x) = Kx$ where $K$ is another constant. Thus all such harmonic functions are of the form $h(x,y) = Kxy + B$.
Hint: if $h$ satisfies your condition, show that $h(x,y)$ is constant on branches of the hyperbolas $xy = c$. Then what does $h(x,y) = f(xy)$ being harmonic tell you about $f$?
