= Spherical Bessel product integral
{title2=$\int_0^\infty j_\ell(kr)j_\ell(kR)\,dk=\frac{\pi}{2(2\ell+1)}\frac{r_<^\ell}{r_>^{\ell+1}}$}
For integer $\ell\geq0$ and $r,R>0$, $r_<=\min(r,R)$ and $r_>=\max(r,R)$. A proof compares the <multipole expansion> of $1/|\mathbf r-\mathbf R|$ with its <Fourier transform of inverse distance in three dimensions>. Applying the <Rayleigh plane-wave expansion> to the Fourier representation gives coefficient $8\int_0^\infty j_\ell(kr)j_\ell(kR)\,dk$ multiplying $Y_{\ell m}(\hat r)Y_{\ell m}^*(\hat R)$; the Coulomb multipole coefficient is $4\pi r_<^\ell/[(2\ell+1)r_>^{\ell+1}]$. Equating the coefficients proves the identity. The powers also match continuously at $r=R$.
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