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?



Fourier transform
First of all observe that


$$ f(x)=\frac{1}{3}\frac{d}{dx} (x^2-1)^{\frac{3}{2}}\_+, $$


where for any real number $t$ we set $t\_+=\max(t,0)$. Thus it suffices to compute the Fourier transform of $(x^2-1)^{\frac{3}{2}}\_+$.


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 $(x^2-1)^\lambda\_+$ 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 $\alpha$. 


