In the future, please post these kinds of questions in the Science Book forum under Academic Guidance.
Arfken, Math Methods for Physicists (2nd ed.), summarizes Fourier transforms (FT's) and works an example on finite wave trains in just 10 pages. Books on quantum mechanics also treat this subject (assuming you already know something about the FT). Merzbacher, QM, 2nd ed., introduces the FT of wave packets in 3 pages (p. 17). You decide how sweet, but you can't get much shorter.