Radiation of a distant point source In class, starting from the Maxwell equations, we showed that :
\begin{equation}
  \mu_0 \vec{J} = (\frac{1}{c^2} \frac{\partial^2{}}{\partial{t^2}} - \vec{\nabla^2})\vec{A}
\end{equation}
Until there, no problem.
Then the teacher said that : 
\begin{equation}
  \vec{A}(\vec{r}, t) = \frac{\mu_0}{4\pi}\iiint_{(\infty)} \frac{\vec{J}(\vec{r}, t - \frac{||\vec{r} - \vec{r}'||}{c})}{||\vec{r} - \vec{r}'||}dV
\end{equation}
My first question is, how can we derive this last expression ?
Then, because the point source is distant,
We showed that : $\frac{1}{||\vec{r} - \vec{r'}||} \approx \frac{1}{r} + \frac{\hat{r} \cdot \vec{r}'}{r^2}$
So that $\vec{A}(\vec{r}, t)$ becomes :
\begin{equation}
  \vec{A}(\vec{r}, t) = \frac{\mu_0}{4\pi} \frac{1}{r} \iiint_{(\infty)} \vec{J}(\vec{r}, t - \frac{r}{c})dV
\end{equation}
I don't really understand how we compute the developpment.
I know we start from computing $||\vec{r} - \vec{r}' ||$ and we ignore the terms of degree 2 or more but I don't end up with same answer.
Thanks for the help.
 A: One way of deriving it is by looking at the solution for Poisson's equation
$$ \nabla^2 \phi = \frac{\rho}{\epsilon_0}  \implies \phi(\mathbf{r}) = -\iiint \frac{1}{4\pi\epsilon_0} \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} d\mathbf{r'}, $$
and introducing a time dependency in the most obvious way, evaluating the source at the retarded time $t_r = t - |\mathbf{r}-\mathbf{r'}|/c$. That this works out is not entirely obvious, and has to be checked by plugging the expression back in the equation. This is how David Griffiths does it in his book "Introduction to Electrodynamics".
In a more general sense, what we are looking for is called a Green's function, that is to say, a solution to the inhomogeneous equation with a delta function on the right-hand side. Because the equation is linear, this allows us to write the solution as a convolution with this Green's function. One way of finding these functions would be to apply a Fourier transform.
Are you sure the last expression is correct? One usually also expands the expression for the retarded time, which in the zero-th order is just the retarded time at the origin
\begin{equation}
  \vec{A}(\vec{r}, t) = \frac{\mu_0}{4\pi} \frac{1}{r} \iiint_{(\infty)} \vec{J}\left(\vec{r}, t - \frac{r}{c}\right)dV
\end{equation}
