I’d like confirmation that
$$ \frac{\cos(b \sqrt{a^2+x^2})}{(a^2+x^2)^{3/2}} $$
has the Fourier cosine transform
$$ \frac{\pi}{2 a} \, \exp(-ay) \qquad \text{if $y>a$,}
$$
as found in *Tables of Integral Transforms* by Arthur Erdelyi et al. equation (35) in Sect. 1.7.

I am puzzled that it is independent of $b$, making me wonder whether the inequality should be $y>b$. Since the numerator is a kind of frequency swing when $x$ is time, I would expect the spectrum to reflect the width $b$ of the swing, (including the case $b=0$) and not its suddenness $a$.

Comparison with equation (29) shows some similarity, but many comparable functions have Fourier transforms with a frequency limit at $b$ rather than $a$. A hint about evaluating the integral might be what I need.

Tables of Integral Transformsis indeed a standard reference, a part of the rather famous Bateman Manuscript Project. $\endgroup$2more comments