How to find the Green's Functions for time-dependent inhomogeneous Klein-Gordon equation? I'm trying to find the Green's functions for time-dependent inhomogeneous Klein-Gordon equation which is : 
\begin{align*}‎‎
\left[ -‎ ‎\nabla ‎^2 + ‎‎‎‎\frac{1}{c^2} ‎‎\dfrac{\partial ^2}{\partial t^2} +‎ ‎‎‎\kappa ‎^2 ‎‎\right] ‎‎\psi(‎{‎\mathbf{r},t )}‎ = ‎‎‎‎\rho‎(‎\mathbf{r},t‎)‎
\end{align*}
It has been mentioned in the question that I can find the Green's functions :
\begin{align*}‎
‎‎&G_R(‎\mathbf{r} , t , ‎\mathbf{r'} , t') = ‎\dfrac{c}{8 \pi ^2 ‎\mathbf{R} i }‎\dfrac{d}{d‎\mathbf{R}} ‎\int_{- ‎\infty‎}^{+‎\infty‎} ‎‎\dfrac{e^{ i ‎\frac{R}{c} ‎‎\sqrt{q^2  - k^2 c^2}‎‎}}{‎\sqrt{q^2  - k^2 c^2}‎} ‎e^{ - iq (t - t')} ‎dq‎
\\‎
&G_A(‎\mathbf{r} , t , ‎\mathbf{r'} , t') = ‎-\dfrac{c}{8 \pi ^2 ‎\mathbf{R} i }‎\dfrac{d}{d‎\mathbf{R}} ‎\int_{- ‎\infty‎}^{+‎\infty‎} ‎‎\dfrac{e^{- i ‎\frac{R}{c} ‎‎\sqrt{q^2  - k^2 c^2}‎‎}}{‎\sqrt{q^2  - k^2 c^2}‎} ‎e^{ - iq (t - t')} ‎dq‎
\end{align*}‎
using the fourier transform, but when I use the fourier transform I don't gain the proper answer. 
The fourier transform which I use is the one which is generally given as  : 
‎\begin{align*}‎
‎f(r) = ‎\dfrac{1}{‎\sqrt{2 \pi}‎} ‎\int_{- \infty}^{\infty} ‎e^{ik.r}‎\hat{f}‎(k) ‎dk ‎‎
\end{align*}
but from this transform I cannot find $G_A$ and $G_R$.
Is there another transform which I should use to find the Green's functions? 
Edit
The Green's function which I wind up with is : 
‎\begin{align*}‎‎
G_A(‎\mathbf{r} , t , ‎\mathbf{r'} , t')‎  = ‎‎‎‎\dfrac{1}{(2\pi)^4} ‎\int ‎d^3\mathbf{k}  ‎dk' ‎‎\frac{1}{k^2} ‎e^{i\mathbf{k}.(‎\mathbf{r} - ‎\mathbf{r'}‎‎)}e^{ik'(t-t')}‎
\end{align*}
which is not even similar to the answer given here!
 A: I'll try to point out things I'm seem that are not fine.


*

*You are making confusion between $k$ and $\kappa$.

*There is no problem with your convention for fourier transform. It's only a little unconventional in this kind of problem (based on the books I used)

*You have to integrate! Note that the integral you found has $d^3{\bf k} dk'$ you must solve it. The common path is to integrate first $k'$, then put the integral in spherical coordinates and solve for the angles. By the end solve the radial integral.


Have you tried first to solve the $\kappa=0$ case? If no, you should do that. 
I'll not reproduce here what's said in some books, instead I'll let you with some sources:


*

*I think the better reference is Hassani book on Mathematical Physics (Sec. 22.4.4), things are clearer there. The problem is that he doesn't solve the $\kappa \neq 0$ case. 

*Another great reference (and which solves your problem) is Eleftherios Economou's book Green's Function on Quantum Mechanics Sec.2.2 (Look page 31, eq 2.63 for your specific case).

*Other references are Bogoliubov Shirkov &15 (which is canonical but a little bit confusing, in my opinion) and 

*Morse Feshbach Chap 7.

