I feel that this "simple" question might have been asked before, but after some searching, I still cannot find it. We often need to know the Fourier transform of the Green's function of $\Delta$ in $\mathbb R^3,$ $$ G(x) =-\frac{1}{4\pi |x|}, $$ which can be written as $$ \hat G(\xi) =-\int_{\mathbb R^3} \frac{e^{ix\cdot \xi}}{4\pi |x|}dx. $$
What are some ways to evaluate this integral? What will the resulting function be like?
Well, $$\Delta G=\delta$$ by definition, so taking the Fourier transform of this gives that $$\hat{G}(\xi)=-(2\pi)^{-3/2}|\xi|^{-2}\in\mathcal{S}'.$$
In fact, $$\hat{G}(\xi)=-(2\pi)^{-n/2}|\xi|^{-2}$$ on $\mathbb{R}^n,$ and this is in $\mathcal{S}'$ for all $n\geq 3$, as $|\xi|^{-2}\in L^1_{loc}$ for such $n$.
Here, I used the convention that $$\hat{u}(\xi)=(2\pi)^{-n/2}\int\limits_{\mathbb{R}^n} u(x)e^{-ix\xi}\, dx.$$