For any linear functional x : R d R x : R d R ,

For any linear functional $x:R^d\to R$, show that

$$\begin{array}{cc}{\int }_{R^d}\frac{\cos \left(\omega x\right)d\omega }{{(1+\parallel \omega {\parallel }^2)}^{v+d/2}}& =I(\nu +1/2,d-1){\int }_R\frac{\cos (w\parallel x\parallel )dw}{{(1+w^2)}^{v+1/2}}\\ =\frac{{\pi }^{d/2}}{2^{v-1}\Gamma (\nu +d/2)}\times \parallel x{\parallel }^v{K}_v(\parallel x\parallel ),\end{array}$$

where $I(v,d)={\pi }^{d/2}\Gamma \left(v\right)/\Gamma (v+d/2)$.