# Integrate Bessel Functions over infinite range

Searching StackExchange doesn't bring up much useful info for this so I'll present you guys with the problem.

I'm trying to replicate the following integrals using Mathematica:

All my best efforts just result with the input being printed out. For example, taking the second integral I try the following:

Assuming[y \[Element] Reals, Integrate[1/Pi BesselK[0, Sqrt[x^2 + y^2]], {x, -Infinity, Infinity}]]


but the result is:

I'm not really sure how to go about gaining the result I need, which should be of the form exp{-ay} as shown in the first image.

Any help much appreciated, thanks.

If you use symmetry to reduce the interval to $[0, \infty)$ and make the substitution $x = \sqrt{y^2 - r^2}$, Integrate can handle the transformed integral. Without loss of generality, one may assume $y > 0$. The substitution, which could be simplified a little but Integrate does not seem to mind, is as follows:

1/Pi BesselK[0, Sqrt[x^2 + y^2]] Dt[x] /.
x -> Sqrt[r^2 - y^2] /. {Dt[r] -> 1, Dt[y] -> 0}
(*  (r BesselK[0, Sqrt[r^2]])/(π Sqrt[r^2 - y^2])  *)


The result:

2 Integrate[
1/Pi BesselK[0, Sqrt[x^2 + y^2]] Dt[x] /.
x -> Sqrt[r^2 - y^2] /. {Dt[r] -> 1, Dt[y] -> 0},
{r, y, Infinity}, Assumptions -> y > 0]
(*  E^-y  *)