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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 $g(\omega)$?

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