Is $ \frac{0.5}{1-(x^2 + y^2)} - \alpha x -\beta y$ with $\alpha, \beta \in \mathbb{R}$, a radial function?
I have to apply some results of radial functions to this one, but it doesn't seem like it's radial at all.
To be clear, meaning that a function $f(x,y)$ is radial if in fact it is actually a function of $r = \sqrt{x^2 + y^2}$ only.
To me, this would imply that since the function only depends on the distance from the origin and not on the angle, the function should evolve in the same way no matter in which direction you look when at the origin.
But after graphing it, the function looks like a tilted plane (except some interesting behavior close to the origin). That doesn't seem to look like a radial function to me... Thoughts?
No, it isn't. The simpe change of variable $$x = \rho \cos \varphi,\\ y = \rho\sin\varphi$$ yields
$$f(\rho,\varphi)=\frac{1}{2(1-\rho^2)}-\rho(\alpha\cos\varphi+\beta\sin \varphi),$$
and the latter parenthesis cannot vanish everywhere unless $\alpha=\beta=0$.