# Fourier transform of $e^{-\delta (x)} \cos\left(\frac{1}{x}\right)$

I have a function

$$f(\omega) = \exp\left(-\frac{\gamma}{\gamma^2+\omega^2}\right)\cos\left(\frac{\omega}{\gamma^2+\omega^2}\right),$$

and I'm trying to calculate its Fourier transform at the limit of $\gamma\rightarrow 0$:

$$\mathcal{F}\left[\lim_{\gamma\rightarrow 0} \, f(\omega)\right].$$

The limit seems to evaluate to

$$g(\omega) = \lim_{\gamma\rightarrow 0} \, f(\omega) = e^{-\pi\delta (\omega)} \cos\left(\frac{1}{\omega}\right).$$

It seems that the Fourier transform of $\cos(\frac{1}{\omega})$ can be related to Bessel functions

$$\mathcal{F}\left[\cos\left(\frac{1}{\omega}\right)\right]=2 \pi \delta(t)-\sqrt{\frac{\pi }{8|t|}}J_1\left(2\sqrt{|t|}\right),$$

but how can I compute the Fourier transform of $g(\omega)$?