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Fourier Series Builder

Learning goal

See Fourier series converge: each added harmonic sharpens the partial sum toward the target waveform, the spectrum bars show which coefficients do the work, and the stubborn overshoot at every jump — the Gibbs phenomenon — shows what convergence "almost everywhere" really means.

Things to try

  1. Square wave, N = 1 → 60. The wiggles crowd toward the jumps but the ~9% overshoot never shrinks. Pointwise vs uniform convergence, live.
  2. Triangle wave. Continuous function ⇒ coefficients fall as \(1/n^2\) ⇒ near-perfect fit by \(N = 10\). Compare the spectrum decay with the square wave's \(1/n\).
  3. Sawtooth — all harmonics present (odd and even). Which symmetry of the square/triangle waves kills their even terms?
  4. Rectangular pulse. The spectrum is a sampled sinc: narrow pulse ⇒ broad spectrum — the uncertainty principle in one picture.

Fourier series · Fourier transform · Fourier integral (eq.)