Hello
I found formula on this book (my book have Date - Edition 1986 year) on page: 218 example: 1.
$$\int_0^{\infty } \exp (-x z) J_0(R x) J_0(r x) \, dx=\Re\left(\frac{Q_{-\frac{1}{2}}\left(\frac{r^2+R^2+z^2}{2 r R}\right)}{\pi \sqrt{r
R}}\right)$$
for: Re[z] > Abs[Im[r]] + Abs[Im[R]]
Numeric check for forumla:
f[R_?NumericQ, r_?NumericQ, z_?NumericQ] := NIntegrate[
Exp[-x z] BesselJ[0, R x] BesselJ[0, r x], {x, 0, Infinity}]; f[20, 2, 1]
(* 0.0500619 *)
g[R_?NumericQ, r_?NumericQ, z_?NumericQ] :=
Re[LegendreQ[-(1/2), (r^2 + R^2 + z^2)/(2 r R)]/(? Sqrt[r R])] // N; g[20, 2, 1]
(* 0.0500619 *)
Regards MI.