How do I find the inverse Hankel transform of $k^2e^{-k^2}$?

Question

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!

Answer 1

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.