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I am trying to solve: $f_l(r)=\int_0^{\infty}e^{-k^2}k^4j_l(kr)dk,$ where $j_l$ is the spherical Bessel function of the first kind, for any integer l >= 0.

Thanks in advance for any answers!

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    Start by looking at $\int_0^{\infty}e^{-s k^2}j_l(kr)dk$ from which you can derive your integral by differentiating twice w.r.t. $s$.2011-04-24

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After searching for a while, I realized there would be no simple expression for this integral. I found the following expressions in Abramowitz and Stegun, which you can download. They involve confluent hypergeometric functions $M(a,b,z)$ for the most general one.

$\int_0^{\infty}e^{-a^2k^2} k^{\mu-1} J_{\nu}(kr)dk = \frac{\Gamma\left(\frac{\mu+\nu}{2}\right)\left(\frac{r}{2a}\right)^{\nu}}{2 a^{\mu} \Gamma(\nu + 1)} M\left(\frac{\mu+\nu}{2},\nu+1,-\frac{r^2}{4 a^2}\right) \; ,$

with $J_{\nu}(z)$ the "ordinary" Bessel function of the first kind. It is related to the spherical one as follows:

$j_{\nu}(z)= \sqrt{\frac{\pi}{2 z}}J_{\nu+1/2}(z) \; .$

If $\mu=\nu+2$, then there is a simpler formula

$\int_0^{\infty}e^{-a^2k^2} k^{\nu+1} J_{\nu}(kr)dk = \frac{r^{\nu}}{(2a^2)^{\nu+1}} e^{-\frac{r^2}{4 a^2}}\; .$

There are some conditions on the range of values for the parameters for these formulae to hold, but I think that should not be a problem in your case. You can find them in the reference as well as further details on confluent hypergeometric functions.

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    Thank you J.M! Using Mathematica I should be able to evaluate a few Ms and then use the recurrence relations in Abramowitz and Stegun to calculate all other Ms.2011-04-25