Riesz transform equivalent form

Question

I learnt from Stein's book that Riesz transform is defined: $\mathcal F(R_j f) = \frac{i\xi_j}{|\xi|} \hat{f}(\xi)$ for $f \in L^2$. I saw a definition: $R_j = \frac{\partial_j}{\sqrt{-\Delta}}$. I don't know how they are related, especially where does the Laplacian operator come from. For $\partial_j$, I guess it comes from $\partial_j |\xi| = \frac{\xi_j}{|\xi|}$. Thanks.