I have a function defined by $$ \psi_{n,\ell,m}(\mathbf{x})=\phi_{n}(x)Y_{\ell m}(\hat{x}) $$
$$ \phi_{n}(x)=\frac{2n^{3/4}\alpha^{3/2}}{\pi^{1/4}}e^{-\frac{n\alpha^2 x^2}{2}} $$
where $\alpha$ is a positive real and $n$ is a positive integer and $Y_{\ell m}(\hat{x})$ is the spherical harmonic. I want to obtain the Fourier Transform of this function, $\widetilde{\psi}_{n,\ell,m}(\mathbf{k})$. Here is the way I was approaching the problem. I wrote
$$ \widetilde{\psi}_{n,\ell,m}(\mathbf{k})=\frac{1}{(2\pi)^{3/2}}\int d^3x e^{i\mathbf{k\cdot x}}\psi_{n,\ell,m}(\mathbf{x}) $$
and from here I used the expression for the exponential
$$ e^{i\mathbf{k\cdot x}}=\sum_{\ell'=0}^{\infty}(2\ell'+1) i^{\ell'}j_{\ell'}(kx)P_{\ell'}(\hat{k}\cdot\hat{x}) $$ where $j_{\ell}(kx)$ are the spherical Bessel functions. This can be further simplified by use of the Addition Theorem relating the Legendre polynomials to spherical harmonics, to ultimately arrive at $$ e^{i\mathbf{k\cdot x}}=\sum_{\ell'=0}^{\infty}\sum_{m'}^{\ell'}4\pi i^{\ell'}j_{\ell'}(kx)Y_{\ell' m'}(\hat{k})Y^{*}_{\ell' m'}(\hat{x}). $$
Plugging this result into the above equation one gets
$$ \widetilde{\psi}_{n,\ell,m}(\mathbf{k})=\frac{1}{(2\pi)^{3/2}}\int d^3x \sum_{\ell'=0}^{\infty}\sum_{m'}^{\ell'}4\pi i^{\ell'}j_{\ell'}(kx)Y_{\ell' m'}(\hat{k})Y^{*}_{\ell' m'}(\hat{x}) \phi_{n}(x)Y_{\ell m}(\hat{x}) $$
The angular integral can be done easily because the two spherical harmonics just set $\int d\Omega_{x}Y^{*}_{\ell' m'}(\hat{x})Y_{\ell m}(\hat{x})=\delta_{\ell' \ell}\delta_{m' m}$, leaving the radial integral
$$ \frac{4\pi}{(2\pi)^{3/2}}\int x^2 dx i^{\ell}j_{\ell}(kx) \phi_{n}(x). $$
This is relatively easy for the case $\ell=0$ case, but for the case $\ell=1$ I get, when doing the integral in Mathematica, a very complicated expression containing the Dawson function in it. That is,
$$ 2\sqrt{2}i\frac{(-k\sqrt{n}\alpha+\sqrt{2}(k^2+n\alpha^2)\text{Dawson}(\frac{k}{\sqrt{2n}\alpha}))}{k^2n^{3/4}\alpha^{3/2}\pi^{1/4}}. $$
I was wondering if there is an easier and more efficient way of computing this Fourier Transform, which can give me the answer in terms of more manageable functions. I need to eventually do integrals in k-space with these functions and as it is they seem too complicated.