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).


[4] <http://www.mathematica-journal.com/2012/12/on-the-integral-of-the-product-of-three-bessel-functions-over-an-infinite-domain/>


[5] [$\mathrm{Bessel}^3$ Integral](https://mathoverflow.net/questions/154382/mathrmbessel3-integral)



An integral involving three Bessel functions
Without loss of generality we can set $c=1$; Mathematica returns a closed form for $a=b$.


For $|a|<1/2$ the result is


$$I= \int\_0^\infty \mathrm{d} x \ x \ J\_0(ax) J\_0(ax) J\_1(x)=\frac{4K(\alpha)}{\pi^2\sqrt{1-4a^2}}\left[2E(\alpha)-K(\alpha)\right]\qquad\qquad(\ast)$$
where $\alpha=\tfrac{1}{2}-\tfrac{1}{2}\sqrt{1-4a^2}$, and $E$ and $K$ are elliptic integrals (as defined [here](http://functions.wolfram.com/EllipticIntegrals/EllipticE/) and [here](http://functions.wolfram.com/EllipticIntegrals/EllipticK/)).


For $|a|>1/2$ Mathematica returns a Meijer-G function,
$$I=\frac{2}{\sqrt\pi} G\_{3,3}^{2,1}\left(\frac{1}{4 a^2}\left|
\begin{array}{c}
 1,1,1 \\
 \frac{1}{2},\frac{3}{2},\frac{1}{2} \\
\end{array}\right.
\right),$$


which equals the real part of $(\ast)$, so I would conclude that the elliptic-integral expression can be used for all $a$.




---


I tried to check this numerically, comparing
$$A=\int\_0^\infty \mathrm{d} x \ x \ J\_0(ax) J\_0(ax) J\_1(x)$$
with
$$B={\rm Re}\,\left[\frac{4K(\alpha)}{\pi^2\sqrt{1-4a^2}}\left[2E(\alpha)-K(\alpha)\right]\right],\;\;\alpha=\tfrac{1}{2}-\tfrac{1}{2}\sqrt{1-4a^2}.$$
The numerical evaluation of $A$ has rapid oscillations for $a<1/2$ as well as for $a>1/2$, and only the envelope is described by $B$. I don't know whether or not this is an artefact of the numerics.  

[Zakk tells me they are an artefact, so it seems the elliptic-integral expression stands: A=B for $a<1/2$ and $a>1/2$.]


![](https://ilorentz.org/beenakker/MO/ellipticintegral.png)  

blue: numerical evaluation of A; gold: plot of B, both as a function of $a$


