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I want to calculate the Fourier Transform of $\hat{f}(s)$ for the function $f(x)=\frac{x}{|x|^3}$ for $x\in\mathbb{R}^3$.

In the application where I am using the Fourier Transform I only care about the case where $s$ is a unit vector, and I'm hoping that, in that case, I'm hoping that $\hat{f}(s)=-i\sqrt{\frac{\pi}{2}}s$ when $|s|=1$ (this will exactly match the coefficient I need in a larger proof where $s$ is a unit normal vector).

I also tried converting the integral into spherical coordinates and my integral became $$\hat{f}(s)=\iiint e_re^{-ir(s\cdot e_r)}\sin(\phi)drd\phi d\theta,$$

where $e_r=(\cos(\theta)\sin(\phi),\sin(\theta)\sin(\phi),\cos(\phi))$

I am not sure how to proceed from here.

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Here is a nice way to proceed rather than trying to compute principal values of oscillations in that integral. Notice that

$$\nabla \cdot \left( \frac{x}{|x|^3}\right) = 4\pi\delta_0$$

as a tempered distribution. Using Fourier transform properties, we have that

$$ik\cdot \hat{f} = 4\pi \hat{\delta_0} = \sqrt{8\pi}$$

using the $\frac{1}{\sqrt{2\pi}}$ prefactor convention for the Fourier transform. Then we have

$$\hat{f}(k) = -i\sqrt{8\pi} \frac{k}{|k|^2}$$

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