TITLE: $\mathrm{Bessel}^3$ Integral
QUESTION [13 upvotes]: 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.

REPLY [18 votes]: 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, 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.