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I would like to carry out symbolically the following integral

$$\int_0^\infty d r \,r^2\, j_0( k r)\, j_0( k_1 r)\, j_0( k_2 r)\,, $$ where $j_0(r)$ is the zeroth order spherical Bessel function and $k$,$k_1$ and $k_2$ are real numbers.

Idea?

I am wondering if I should use this expansion

$$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m}$$

from this reference

Clue

If I am to believe Mathematica $$\int_0^\infty d r \,r^2\, j_0( r)\, j_0( 2 r)\, j_0( 3 r)=\frac{\pi}{48}$$ for instance, so the integral seems possible.

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1 Answer

up vote 2 down vote accepted

$$ \begin{align} &\int_0^\infty\mathrm drr^2j_0(k_1r)j_0(k_2r)j_0(k_3r) \\ =& \int_0^\infty\mathrm dr\frac{\sin k_1r\sin k_2r\sin k_3r}{k_1k_2k_3r} \\ =& \int_0^\infty\mathrm dr\frac{\sin(k_1+k_2-k_3)r+\sin(k_2+k_3-k_1)r+\sin(k_3+k_1-k_2)r-\sin(k_1+k_2+k_3)r}{4k_1k_2k_3r} \\ =& \frac{\pi\left(\def\sgn{\operatorname{sgn}}\sgn(k_1+k_2-k_3)+\sgn(k_2+k_3-k_1)+\sgn(k_3+k_1-k_2)-\sgn(k_1+k_2+k_3)\right)}{8k_1k_2k_3}\;. \end{align} $$

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Thanks a lot! It always looks so simple in retrospect; I feel a bit embarrassed. –  chris Nov 27 '12 at 9:32
    
@chris: You're welcome! –  joriki Nov 27 '12 at 9:33
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