Fourier Series
Any periodic function = sum of sinusoidal harmonics. Watch epicycles compose waves in real time.
π΅ Fourier Series
Jean-Baptiste Joseph Fourier showed (1822) that any periodic function f(x) can be written as an infinite sum of sines and cosines:
f(x) = aβ/2 + Ξ£βββ [aβ cos(nx) + bβ sin(nx)]
Coefficients are computed by projection: aβ = (1/Ο) β« f(x) cos(nx) dx bβ = (1/Ο) β« f(x) sin(nx) dx
Gibbs phenomenon: near jump discontinuities (square/sawtooth waves) the partial sum overshoots by approximately 8.9% no matter how many terms are added β a fundamental property of Fourier approximation near discontinuities.
Parseval's theorem: the total energy in a signal equals the sum of energies in each harmonic: (1/Ο) β« |f|Β² dx = aβΒ²/2 + Ξ£ (aβΒ² + bβΒ²)
The epicycle diagram (left) shows each harmonic as a rotating arm β the tip traces the function value, just as Ptolemy's epicycles approximated planetary paths.