A simple series-expansion form for the direct-scattering amplitude (involving real quantities) T n 1 s - n 2 s d is given in the case of the n 1 s - n 2 s transition in the electron—hydrogen-atom collision. Another form for T n 1 s - n 2 s d , consisting of few terms, is derived when the initial-state quantum number n 1 is small. When the quantum numbers n 1 and n 2 are very large, the dominant part of T n 1 s - n 2 s d can be reduced to a single oscillatory trigonometric term, and this leads to some interesting features of the differential cross section. Simple integral forms for T n 1 s - n 2 s d in the above-mentioned cases are also given. Expressions are further obtained for the exchange-scattering amplitude T n 1 s - n 2 s ex , which includes terms of orders O ( β 0 - 1 ) and O ( β 0 - 2 ) , with β 0 = k 1 2 + n 2 - 2 = k 2 2 + n 1 - 2 , where k 1 ( k 2 ) is the initial (final) electron momentum. We also derive exact expressions for T 2 s - 2 s ex and T 1 s - 3 s ex . All calculations are done in the first Born approximation.
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