I got a function $f(x) = \frac {1}{4\pi x^2 + k^2}$, where $k \gt 0$ is a constant. And I wanted do make the fourier transform in $\Bbb R^3$ (it is solved in my textbook). But in the solution it was said that the transform is in the sense of the distributions. But why? Why this function isn't Lebesgue integrable, in $L^1(\Bbb R^3)$?
I understand that $\frac {1}{x^2}$ isn't in $L^1$, because the function isn't continuous in $x=0$. Even though $\frac {1}{x^2}$ goes to zero for $x \to \infty$. But I don't see any discontinuity in $\frac {1}{4\pi x^2 + k^2}$. So why it is needed to use the radial distribution. Because of the dimension? I am confused.