Fourier series and transforms
Chapters 5 and 15 of Nearing's text, Chapters 14 and 15 of Arfken & Weber.
Basics of Fourier series (where/how they arise), choice of basis
functions. Fourier transforms: as extensions of Fourier series,
relation to the Laplace transform. Useful properties of Fourier
transforms (convolution, Parseval's theorem, Weiner-Khinchin theorem),
common Fourier transform pairs, Green's functions for differential equations
Applications
(transfer functions, image processing/Fourier optics, wave packets in
quantum mechanics, time series analysis/musical notes), etc.