Taking the Fourier transform of a Hankel function

Considering the following inverse Fourier transform

$$f(t) = -\alpha \int_{-\infty}^{\infty} F(\omega)H_0^{(2)}(k(\omega) \beta) \exp(+j\omega t) d\omega$$

where $F$ is an arbitrary function and $H_0^{(2)}$ is the Hankel function of second kind and order zero, and $\alpha, \beta$ are constant.

The main question is; how to evaluate the (inverse) Fourier transform of a Hankel function ?

I've been told that in order to evaluate this analytically almost everything requires approximations. Therefore, it is computed numerically in general.

Any suggestions on what method that should be used and why ?

Kind regards.