I have evaluated the following integration using Mathematica. I obtained a solution in terms of Meijer G function. I wonder if it can be simplified to be in terms of Bessel functions.
FullSimplify[
Integrate[
x Exp[-θ x^2/(
4 t) ] (BesselJ[0, λ x] BesselY[0, λ] -
BesselY[0, λ x] BesselJ[0, λ]), {x, 0,
Infinity}]]
(2 t (E^(-((t λ^2)/θ)) BesselY[0, λ] -
BesselJ[0, λ] MeijerG[{{0}, {-(1/2)}}, {{0,
0}, {-(1/2)}}, (t λ^2)/θ]))/θ
FunctionExpand[]
on it? $\endgroup$ – J. M.'s ennui♦ Jan 12 at 0:02theta
,lambda
, andt
? $\endgroup$ – Bob Hanlon Jan 12 at 1:19