I would like to understand if there is a closed formula for this integral:
$$\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,\frac{J_1(kR)}{k^2}$$
where $R,\delta>0$ and $J_1(\cdot)$ is the bessel function of first kind of order 1.
By using the series expansion of $J_1$ and the Guassian integral, I end up with this series (with a prefactor $\delta$):
$$\sum_{l=0}^{\infty}\frac{(-1)^l}{2^{2l+2}}\,\frac{(l-1)!}{l! (l+1)!}\,\left(\frac{R}{\delta}\right)^{2l+1}$$
Checking with Wolfram the series does converge but is it possible to find a nicer solution? Or otherwise another way to solve the integral above?
With the help of Mathematica I got:
$$\mathcal{L}\left(\frac{J_1(R\sqrt{x})}{x^{3/2}}\right)=\frac{|R|}{2}\left(1-2\gamma-\frac{4s}{R^2}\left(1-e^{-\frac{R^2}{4s}}\right)+\log\frac{4}{R^2}-\Gamma\left(0,\frac{R^2}{4s}\right)\right)$$ and by expanding the RHS as a series it is not difficult to check it matches your series.