Does anyone know what the Fourier transform (in the sense of distributions) of $$ f(x) = (x^2  1)^{1/2}x, \quad x\ge 1, $$ and $f(x) = 0$ otherwise, is?
First of all observe that $$ f(x)=\frac{1}{3}\frac{d}{dx} (x^21)^{\frac{3}{2}}_+, $$ where for any real number $t$ we set $t_+=\max(t,0)$. Thus it suffices to compute the Fourier transform of $(x^21)^{\frac{3}{2}}_+$. In Section 2.5 Chapter 2 of the book by Gelfand and Shilov, Generalized Functions, vol.1, Academic Press 1964, the authors compute the Fourier transform of $(ax^2+bx+c)^\lambda_+$. Your example corresponds to Case (3) discussed there. More precisely the Fourier transform of $(x^21)^\lambda_+$ is the function $$\Gamma(\lambda+1)\sqrt{\pi}\left\frac{\xi}{2}\right^{\lambda\frac{1}{2}}\frac{\cos\pi(\lambda+\frac{1}{2}) J_{\lambda\frac{1}{2}}(\xi)J_{\lambda+\frac{1}{2}}(\xi)}{\sin \pi(\lambda+\frac{1}{2})}, $$ where $J_\alpha$ denotes the Bessel function of order $\alpha$. 

