I have an integral transform (motivated by a physics problem)
$F (x) = \int_0^\infty dx' \frac {a x'} {(a x)^2 + (x - x')^2} f (x')$,
where $x, x', a > 0$ real, $f : \mathbb{R}_+^0 \to \mathbb{R}$.
Is this this transform known and classified in any manner? And if yes, does there exist an inverse, so that we can obtain $f (x')$ unknown from $F (x)$ known
$f (x') = \int_0^\infty dx K (x', x) F (x), F : \mathbb{R}_+^0 \to \mathbb{R}$?