Properties of the Fourier transform of a certain function

Question

In my research I met the Fourier transform of the function $f(x)=(1+x^2)^{-1/2}$. I was not able to find its explicit formula. Is this a function known as a 'special function'? I would like to know if it is nonnegative, summable, etc.

Answer 1

If the fourier transform is defined by $\;\displaystyle\int_{-\infty}^\infty f(x)e^{-2\pi ikx}\,dx\;$ then the answer is $$\sqrt{\frac 2{\pi}}K_0(|t|)$$

as provided by Alpha or in the Wolfram functions 'transforms' of the modified Bessel $K$ function.

This definition from DLMF (and all information there !) may be the most appropriate : $$K_0(x)=\int_0^\infty \cos(x\;\sinh(t))dt=\int_0^\infty \frac{\cos(x\;t)}{\sqrt{1+t^2}}dt$$ (remembering that $\;\displaystyle\operatorname{argsinh}'(x)=\frac 1{\sqrt{1+x^2}}\ $)