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Harmonic functions

Harmonic functions. Sines and cosines with the same frequency. The general picture: Asin  x + Bcos  x. (x is usually t). I’ll set A = 1 and B = 0 for examples here. Examples with 0 ≤ x ≤ 1000,  = 2nπ/1000. Sums of harmonic functions with different frequencies are not harmonic.

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Harmonic functions

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  1. Harmonic functions Sines and cosines with the same frequency The general picture: Asinx + Bcos x (x is usually t) I’ll set A = 1 and B = 0 for examples here. Examples with 0 ≤ x ≤ 1000,  = 2nπ/1000

  2. Sums of harmonic functions with different frequencies are not harmonic.

  3. That one was periodic because the n = 6 has the same period as the n = 3 Incommensurate periods give apparently aperiodic sums

  4. But almost all sums of harmonic functions are periodic And almost all periodic functions are sums of harmonic functions (albeit possibly infinite sums) (Can we say “Fourier series” boys and girls? We’ll look at that this evening.)

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