If $W=\frac{1}{4\pi|x|}$ is the Coulomb potential and we define the energy to be $E=-\nabla_xW*f$, then I am trying to prove that $\widehat E=i\frac{k}{|k|^2}\widehat{f}$ on the Fourier side with convention $$\widehat{f}(k)=\int_{\mathbb R^3}e^{-ik\cdot x}f(x)\,dx.$$
But I compute $\widehat{-\nabla_xW*f}(k)=-(ik)\widehat{W}(k)\widehat f(k)=-\frac{ik}{|k|^2}\widehat f(k)$ but I am out by a factor of $-1$ as the book says that it's $+\frac{ik}{|k|^2}\widehat f(k).$