I am looking for a closed form for the following integral

$$ I = \int_0^\infty \mathrm{d} x \ x \ J_0(ax) J_0(bx) J_1(cx) $$

which can be thought of as a particular case of the more general integral

$$ I(n_1,n_2,n_3) = \int_0^\infty \mathrm{d} x \ x \ J_{n_1}(ax) J_{n_2} (bx) J_{n_3} (cx) $$

i.e. $I=I(0,0,1)$. I am aware of the following related results:

- A closed form, analytic result for I(0,0,0), see for instance Ref. [1].
- A closed form, analytic result for I(0,1,1), also in Ref. [1].
- A closed form, analytic result for the general case $I(n_1,n_2,n_3)$ when $n_1 + n_2 + n_3 = 0$, in Ref. [2].
- A general expression in terms of the hypergeometric function $F_4$, which would include I(0,0,1) as a particular case, which however requires $c>a+b$. This is in Ref. [3].
- The analytic/numerical approach of Ref. [4]

Still I cannot find a result for $I=I(0,0,1)$ for every value of $a$,$b$,$c$. I have verified numerically that the integral exists finite, even when the condition $c>a+b$ is not met.

So far I have tried expressing $J_1(cx)$ using recursion relations to try and recover other known integrals, or integrating by parts one or more Bessel functions, or expressing the Bessel functions as a series. All these approaches seems to make the integral more complicated.

Similarly to this [5] other question, the motivation is coupling of momenta in Quantum Mechanics.

[1] G.N. Watson, "A Treatise on the Theory of Bessel Functions", (Cambridge University Press, Cambridge), 1966.

[2] A. D. Jackson and L. C. Maximon, "Integrals of Products of Bessel Functions", SIAM J. Math. Anal. 3, 446 (1971).

[3] W.N. Bailey, "Some Infinite Integrals Involving Bessel Functions", Proceedings of the London Mathematical Society 40, 37 (1936).

Mean Motionpaper mentioned by @AlexandreEremenko. Do you think you can get something useful on $\int_0^\infty x\,J_0(ax)\,J_0(bx)\,J_1(cx)\,dx$ from this? $\endgroup$ – Gro-Tsen May 19 '18 at 11:57