I will link the following question, as it is partly related to the problem I am trying to deal with.
Green's function for the inhomogenous Klein-Gordon equation
As you can read from this User´s question, they are trying to solve the Inhomogenous Klein-Gordon equation. They do this by first looking up for a Bessel representation of the corresponding Green function, which they would then plug into an integral to get the solution. All the referred equations are in their question. My question, however, concerns the derivation of the Bessel representation they found.
I tried deriving the corresponding Green function by looking at different papers. I am considering the (1+1)-dimensional case, I got to the following result for the retarded propagator:
$$G(x-x´, t-t´) = \theta(t-t´)\int \frac{dk}{2\pi} \frac{\sin(w_k(t-t´))}{w_k}e^{ik(x-x´)} \quad\text{,}\quad w_k = \sqrt{k^2+m^2}$$
where $k$ is the wavenumber and $w$ is the frequency. Now, various papers report that, for the (1+1)-dimensional problem, the corresponding Bessel representation is:
$$G(x-x´, t-t´) = \theta(t-t´)\frac{J_0(m\tau)}{2} \quad\text{,}\quad \tau= \sqrt{(t-t´)^2-(x-x´)^2}$$
where $J_0(\dot)$ is the zeroth order Bessel function. My question is: How do we go from the first to the second representation? How would one derive Bessel´s function from that integral?
