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  *)

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