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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
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  • $\begingroup$ Have you tried using FunctionExpand[] on it? $\endgroup$
    – J. M. can't deal with it ♦
    Jan 12 '21 at 0:02
  • $\begingroup$ Yes. I have. But it doesn't simplify it. $\endgroup$
    – Refaat Galal
    Jan 12 '21 at 0:10
  • $\begingroup$ Then it looks like Mathematica is unaware if the $G$-function has a further simplification, if it does have one. $\endgroup$
    – J. M. can't deal with it ♦
    Jan 12 '21 at 0:17
  • $\begingroup$ What constraints/assumptions, if any, exist for parameters theta, lambda, and t? $\endgroup$
    – Bob Hanlon
    Jan 12 '21 at 1:19
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I think that your Meijer G function is equal to $\frac{e^{-z}\operatorname{Ei}(z)}{\pi}$. I cannot prove it except plotting their difference to ridiculous precision, and referring to this similar-looking identity on the Wolfram Functions site:

g[z_] = MeijerG[{{0}, {-(1/2)}}, {{0, 0}, {-(1/2)}}, z];
h[z_] = E^(-z) * ExpIntegralEi[z] / π;

Plot[g[z] - h[z], {z, 0, 2}, WorkingPrecision -> 10^4]

enter image description here

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    $\begingroup$ Evaluating DifferentialRootReduce[MeijerG[{{0}, {-1/2}}, {{0, 0}, {-1/2}}, z] - Exp[-z] ExpIntegralEi[z]/Pi, z] yields a 4th-order ODE with all initial conditions equal to 0, so... $\endgroup$
    – J. M. can't deal with it ♦
    Jan 12 '21 at 9:17
  • $\begingroup$ Thanks for your response. I reached to the same result by substituting numerical values for theta, lamda, and t. Thanks again. @Roman $\endgroup$
    – Refaat Galal
    Jan 12 '21 at 15:07

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