Determine if the function $f(x,y) = \ln(\|(x,y)\|)$ is harmonic in some subset of the plane.
So knowing that from the Laplacian $$\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} = 0 $$
I could determine if its harmonic I went to compute the second-order partials and got that
$$\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} = \frac{y^2-x^2}{(x^2+y^2)^2} - \frac{x^2-y^2}{(x^2+y^2)^2} = 0 $$
This resulted in $2y^2-2x^2=0$
I'm not sure I follow how I can find the subset of the plane here? Restricting $x$ and $y$ somehow?