I have to work out the integral:

$$ \int_0^{\infty} dq \frac{J_0(q \xi)}{q+1} $$

where $ J_0(z) $ is the Bessel function of the first type and order zero, $\xi \in \mathbb{R}$, $\xi \ge 0 $. So far, I only collected a good amount of trials and errors.

This problem actually comes from physics. I am Fourier transforming the 2-dimensional Coulomb potential $W(q)$ to real space coordinates, when screening effects in the the electron gas are accounted for as in the RPA, in the static and long wavelength limit so that:

$$ W(q) = \frac{2\pi e^2}{A} \frac{1}{q + \Lambda} $$

where $e$ is the electron charge, $A$ is the area, $q$ the modulus of wavevector $\mathbf{q}$ and $\Lambda$ (real positive constant) the screening wavevector. When transforming to real space coordinates $\mathbf{x}$, one arrives at the integral above (unless I made some mistakes), with $\xi = \Lambda x$, $x$ being the modulus of $\mathbf{x}$. So it would be physically interesting to have the large $x$ limit and small $x$ behaviour.