# How to simplify the following integral to be in terms of Bessel functions?

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

• Have you tried using FunctionExpand[] on it? – J. M.'s ennui Jan 12 at 0:02
• Yes. I have. But it doesn't simplify it. – Refaat Galal Jan 12 at 0:10
• Then it looks like Mathematica is unaware if the $G$-function has a further simplification, if it does have one. – J. M.'s ennui Jan 12 at 0:17
• What constraints/assumptions, if any, exist for parameters theta, lambda, and t? – Bob Hanlon Jan 12 at 1:19

## 1 Answer

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] • 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... – J. M.'s ennui Jan 12 at 9:17
• Thanks for your response. I reached to the same result by substituting numerical values for theta, lamda, and t. Thanks again. @Roman – Refaat Galal Jan 12 at 15:07