Scalar particle scattering amplitude

[faklif]

Homework Statement

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

Now in the same text I have the potential $$A^0 = \frac{Ze}{4\pi |x|}$$ where x are the spatial coordinates and $$A^i = 0$$. 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.