Example of a simple step function on (0,1). Extended it to make a Fourier sin series, cos series (each with period 2), and a complex exponential series with period 1.
Discussed interpretation of Fourier coefficients in terms of what you hear (e.g.)
Introduction to Fourier transforms - needed if your function is not at all periodic, and/or if you need a continous spectrum of frequencies. (Same thing!)
Set up formalism - what happens in the limit that period goes to infinity?
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