I throw this integration into Mathematica,
Integrate[ (z BesselJ[0, s z])/(z^2 - w^2), {z, 0, [Infinity]}]
and get the result as,
ConditionalExpression[BesselK[0, s/Sqrt[-(1/w^2)]], Re[s] > 0 && Im[s] == 0 && Im[w] != 0]
So it is basically telling me that, for real s, approaching the limit $w\rightarrow1+i0^+$,
$$\int_0^{+\infty}dz \frac{z J_0(s z)}{z^2-1}=K_0(-i s)$$
which I have a hard time trying to replicate by hand.
Is the above expression correct? Can you show me the magic therein?
relevant: Integration of Bessel functions:Finding a suitable contour