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!
