Scalar particle scattering amplitude

[faklif]

Homework Statement

I have simplified the expression
[tex]-i\int d^4xie[\phi'^*(\partial_{\mu}A^\mu + A^\mu\partial_\mu)]\phi[/tex]
to
[tex]-i\int d^4xie[\phi'^*(\partial_\mu\phi) - (\partial_\mu\phi'^*)\phi]A^\mu[/tex]
under the conditions
[tex]A^0 \rightarrow 0, t \rightarrow \pm \infty[/tex]
[tex]|A^i| \rightarrow 0, |x_i| \rightarrow \infty[/tex]
using partial integration. As was requested in a problem in the text.

Now in the same text I have the potential [tex]A^0 = \frac{Ze}{4\pi |x|}[/tex] where x are the spatial coordinates and [tex]A^i = 0[/tex]. My problem is that the book uses the above results which I derived under conditions which I don't see that this potential meets.

Homework Equations

See above.

The Attempt at a Solution

 

[NateD]

I'm assuming that I've done something wrong because the conditions in my simplification don't match the potential given. But I can't see what.