In this paper, page 252, there is an equality (D.69), the numerical results integrals on the left is of interest:
$$\int_{-\infty}^{\infty}\dfrac{x\cdot J_0(x\cdot r)}{x^2-k^2}dx=\left\{ \begin{array}{cc} \dfrac{i \cdot\pi \cdot H_0^{(1)}(k\cdot r)}{2}&\text{Im}(k)>0 \\ \\ \dfrac{i \cdot\pi \cdot H_0^{(2)}(k\cdot r)}{2} &\text{Im}(k)<0\\ \end{array} \right. \tag{D.69}$$
Where $J_0$ is the BesselJ, $H_0$'s are the Hankel functions respectively.
I don't know how to prove the integral equality or whether it holds only for the non-real $k$'s .
But I am interested in the numerical (or analytical when possible) result when $k$ is a real positive number.
But how to obtain a numerical solution of such integrals? e.g., I tried to let $k=10$ and $r=6$, but found most of the available integral algorithms in NIntegrate didnot lead to easy convergence probably due to the existence of a discontinuity point $x=k$ of the integrand and the improper upper limit.
e.g..
NIntegrate[(x BesselJ[0, 6 x])/(x^2 - 100), {x, 0, Infinity}, Method -> {"OscillatorySelection", "TermwiseOscillatory" -> True, Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 10000}}, MaxRecursion -> 100, PrecisionGoal -> 8, AccuracyGoal -> 5] gives
0.155652
and warning messages:
NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in x near {x} = {10.0009}. NIntegrate obtained 0.16591870616051807
and 0.2705956661691797for the integral and error estimates.
Integrate[(x BesselJ[0, 6 x])/(x^2 - 100), {x, -Infinity, Infinity}]$\endgroup$