This note derives the Born differential cross section of addendum {A.30}.
The general idea is to approximate (A.228) for large distances . Then the asymptotic constant in (A.216) can be identified, which gives the differential cross section according to (A.218). Note that the Born approximation took the asymptotic constant equal to one for simplicity.
The main difficulty in approximating (A.228) for large
distances is the argument of the exponential in the fraction. It
is not accurate enough to just say that is
approximately equal to . You need the more accurate
approximation
It follows that