So, I've been scratching my head over this for the whole day. I'm trying to solve the following integral
$$\int^\infty_{-\infty} \frac{e^{i \alpha (X-\xi)}}{\sqrt{\alpha^2+ \beta^2 }} \, d\alpha$$
$\beta$ here is just a constant. So just for some context, I took a Fourier transformation in order to solve a partial differential equation and after solving and using boundary conditions I obtained an expression in Fourier Space. I am now trying to find the Inverse Fourier transform, which gives the above integral which I can not solve.
I've solved the following using contour integration
$$\int^\infty_{-\infty} e^{i \alpha (X-\xi)} (i \alpha)^{3/4} \, d\alpha$$
Thanks guys.
The Fourier transform of $\frac{1}{\sqrt{x^2+\beta^2}}$ is a modified Bessel function of the second kind. For instance,
$$\mathcal{F}\left(\frac{1}{\sqrt{x^2+1}}\right) = \sqrt{\frac{2}{\pi}}\,K_0(|s|).\tag{1}$$
Not by chance, the RHS of $(1)$ is the density function of a normal product distribution.