The Fourier Series
The Fourier Series operates on a time signal that is periodic, i.e., a time signal whose waveform repeats over and over again out to infinite time. Fourier showed that such a signal is equivalent to a collection of sine and cosine functions whose frequencies are multiples of the reciprocal of the period of the time signal. The rather unexpected result is that any wave shape whatsoever, as long as it is not infinite in length, can be represented, as the sum of a collection of harmonic components, and the fundamental frequency of the harmonic series is 1 divided by the length of the wave shape. The amplitudes of the various harmonics are called the Fourier coefficients, and their values can be calculated easily if the equation for the wave shape is known. They can also be calculated graphically from the wave shape itself. A certain physics class is known to have done this with the silhouette of Marilyn Monroe. They posted the MM coefficients on the bulletin board as an "in" joke.
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