Far field approximation for massive Klein-Gordon equation in 3+1D For a massless scalar, one has the familiar Green's function
$$
G(t,r) = \frac{\delta(t - r)}{4\pi r}\,,
$$
and one may take the far-field approximation in a rather straight-forward way:
$$
\int d t d^3 x G(t - t', |\vec x - \vec x'|)f(t,\vec x)\approx \frac{1}{4\pi r'}\int dtd^3 x \delta(t - t' - r + r'(\hat x\cdot\hat x'))f(t,\vec x)\,.
$$
However, in the case of a massive scalar, the Green's function gets a bit uglier
$$
G\supset \frac{\mu}{4\pi \sqrt{t^2 - r^2}}J_1(\mu \sqrt{t^2 - r^2})\,.
$$
In order to do any sort of analytical simplification, I'd like to get rid of the roots, or any other unwieldy powers of the coordinates. If I attempt to take the far-field approximation, which amounts to $r'\gg r$ (and consequently $t'\gg t$), I obtain
$$
\sqrt{(t - t')^2 - |\vec x - \vec x'|^2}\to \sqrt{t^2 - r^2} - \frac{t t' - \vec x\cdot\vec x'}{\sqrt{t^2 - r^2}}\,.
$$
However, one immediately sees that if $r\to t$, the approximation breaks down. Ultimately, I'd like to take the $t,r\to\infty$ limits, and I would suppose that by first taking $t\to\infty$ and then $r\to\infty$ I could avoid this problem.
With all this said, I am getting myself very confused about orders of limits, etc. Is there an example where someone has explicitly worked something like this out? Do any of you have experience with doing integrals of this form?
 A: You don't need to take $t' \rightarrow +\infty$ and $r' \rightarrow +\infty$ individually. The quantity that should be large is the proper distance i.e. $\sqrt{t' ^{2} - r' ^{2}} \rightarrow +\infty$. The expansion of the root then becomes as you wrote it (except for the primed quantities being in the argument of the roots).
With this expansion, the Bessel function now has the form $J_{1}\left [\mu \left (a - \frac{x}{a} \right ) \right ]$ where $a = \sqrt{t' ^{2} - r' ^{2}}$ your very large constant and $x = t't - \mathbf{r'} \cdot \mathbf{r}$ your variable. Taylor expanding about $x = 0$ and keeping the leading and two subleading terms you get:
\begin{equation}
J_{1}\left [\mu \left (a - \frac{x}{a} \right ) \right ] \approx J_{1}(\mu a) + x \frac{\mu J_{2} (\mu a)}{a} - x \frac{J_{1}(\mu a)}{a^{2}}
\end{equation}
Taylor expanding the inverse of the square root as you correctly reproduced it will also yield:
\begin{equation}
\frac{1}{\sqrt{(t' -t)^{2} - |\mathbf{r'} - \mathbf{r}|^{2}}} \approx \frac{1}{\sqrt{t'^{2} - r'^{2}}} \left ( 1 + \frac{t't - \mathbf{r'} \cdot \mathbf{r}}{t'^{2} - r'^{2}} \right )
\end{equation}
So in the end the Green's function will have the following behaviour:
\begin{equation}
G \sim \frac{\mu}{\sqrt{t'^{2} - r'^{2}}} \left ( 1 + \frac{t't - \mathbf{r'} \cdot \mathbf{r}}{t'^{2} - r'^{2}} \right ) \, \left [ J_{1}(\mu \sqrt{t'^{2} - r'^{2}}) +  (t't - \mathbf{r'} \cdot \mathbf{r}) \frac{\mu J_{2} (\mu \sqrt{t'^{2} - r'^{2}})}{\sqrt{t'^{2} - r'^{2}}} - (t't - \mathbf{r'} \cdot \mathbf{r}) \frac{J_{1}(\mu \sqrt{t'^{2} - r'^{2}})}{t'^{2} - r'^{2}} \right ]
\end{equation}
I'd advise to keep all of these terms in the integral (since now it becomes quite doable) and making any further approximations afterwards.
