I am trying to compute the 3D Fourier transform of a spherically symmetric function of the form $$f(r) = e^{\frac{1}{r} e^{-r}} - 1\, ,$$ which entails the integral $$\begin{aligned}F(k) =& \int d^3\!r \, e^{-i\mathbf{k}\cdot\mathbf{r}} f(r) \\ =& \int_0^\infty dr\!\int_0^{\pi}\!d\theta\!\int_0^{2\pi}\!d\phi \, r^2 \sin\theta \, e^{-ikr\cos{\theta}} f(r) \\ = & \, 2 \pi \int_0^\infty\! dr \, r^2 f(r) \int_0^{\pi}\!d\theta \sin\theta \, e^{-ikr\cos{\theta}} \\ =& \, 4\pi \int_0^\infty\!\! dr \, r^2 \mathrm{sinc}(kr) \, f(r) \,.\end{aligned} $$ Based on the answer to this question, I'm inclined to think that this integral will not converge even if I attempt to regularize it because exponent grows without bound near $r=0$. However, since the integral is not a usual Fourier transform, I'm holding out hope that the integral converges, though by what means, I do not know. My main questions are then:
- Is the reasoning within the linked answer applicable to this "3D" Fourier transform? That is, does this spherically symmetric function follow the same convergence rules as that of a one-dimensional cartesian-coordinate function?
- If the link answer does not apply here, what tools are available to try to compute this transform? I can readily accept a good approximation, as this is with regards to a physics problem.