I'm trying to calculate the following integral:
$\int\_0^\infty \mathrm{BesselJ}[l\_0,k\_0r] \cdot \mathrm{BesselJ}[l\_1,k\_1r] \cdot \mathrm{BesselJ}[l\_0-l\_1,kr] \cdot r\,dr$


($\mathrm{BesselJ}[n,x]$ is the Bessel function of the first kind of order $n$)
I assume that $l\_0,l\_1$ are integers, and that $k\_0,k\_1,k>0$.


The result has nice implications in quantum mechanics - to explain selection rules.


I was able to prove that the integral is zero when $k,k\_0,k\_1$ cannot be the lengths of a triangle.
So there is a non-zero result when $|k\_0-k\_1| < k < k\_0+k\_1$
However, I don't know how to calculate the integral when it is not zero.
Can anyone help?


I tried using Mathematica to get a numerical answer (which is fine for me), but I do not think I can count on it. Mathematica is giving a non-zero result for the regime where the integral should be zero.



$\mathrm{Bessel}^3$ Integral
In fact, precisely your integral has been computed in closed form, in: Annie Gervois and Henri Navelet, [*Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities*](http://www.ams.org/mathscinet-getitem?mr=761861), J. Math. Phys. 25 (1984), no. 11, 3350–3356. Their result is
$$
\int\_0^\infty J\_m(ar)J\_n(br)J\_{m+n}(cr)r\,dr =
\begin{cases}
0&\text{if }c^2 < (a-b)^2\\
\dfrac{\cos(mB-nA)}{\pi ab\sin C}&\text{if }c^2 = a^2+b^2-2ab\cos C\\
0&\text{if }c^2 > (a+b)^2
\end{cases}
$$
where in the second case, $A, B, C$ are the angles opposite to sides $a, b, c$ of the resulting triangle. (They give formulas valid for real $m, n$, which reduce to the above when $m, n\in\mathbf Z$ as you assume.)


To put this in your notation, take $(m,n,a,b,c)=(l\_0,-l\_1,k\_0,k\_1,k)$ and use that $A+B+C=\pi$ and $J\_n=(-1)^nJ\_{-n}$. We get
$$
\int\_0^\infty J\_{l\_0}(k\_0r)J\_{l\_1}(k\_1r)J\_{l\_0-l\_1}(kr)r\,dr =
\begin{cases}
0&\text{if }k < |k\_0-k\_1|\\
\dfrac{\cos([l\_0-l\_1]K\_1-l\_1K)}{\pi k\_0k\_1\sin K}&\text{if }|k\_0-k\_1|<k<|k\_0+k\_1| \\
0&\text{if }k > |k\_0+k\_1|
\end{cases}
$$
where $K\_0,K\_1,K$ are the angles opposite to $k\_0,k\_1,k$. The denominator is $2\pi$ times the triangle's area.


