This integral does not converge, how is it solved that it converges in the sense of distribution? if there is, thank you.
This integral doesn't converge absolutely, but it does converge for most values of $a$. First, suppose that $a=b^2>0$ let us note that it is well known that
where $K_0(x)$ is a modified Bessel function. It is obvious that the integral diverges when $b\to 0$. We can produce the integral in question from this one by taking a derivative with respect to $x$ which gives
which again converges just fine for any non-zero values of $x,b$.
Also the integral converges for $a<0$ since the branch points on the real axis are integrable.