y=sin⁡x+sin⁡2x2+sin⁡3x3y = \sin x + \dfrac{\sin 2x}{2} + \dfrac{\sin 3x}{3}

A Fourier Partial Sum y=sin⁡x+sin⁡2x2+sin⁡3x3y = \sin x + \dfrac{\sin 2x}{2} + \dfrac{\sin 3x}{3}

y=sin⁡x+sin⁡2x2+sin⁡3x3y = \sin x + \dfrac{\sin 2x}{2} + \dfrac{\sin 3x}{3} is the infinite series ∑n=1∞sin⁡nxn\sum_{n=1}^{\infty}\dfrac{\sin nx}{n} cut off after its first three terms1. It is a function for watching how adding a few smooth sine waves brings out an angular, discontinuous waveform.

Domain and symmetry

The domain is all real numbers. Each term is odd, so the sum is odd and has point symmetry about the origin. The period is 2π2\pi, and at x=0x = 0 and x=πx = \pi every term vanishes, so y=0y = 0.

What it approaches

The limit of the full sum is the following line on 0<x<2π0 < x < 2\pi.

∑n=1∞sin⁡nxn=π−x2\sum_{n=1}^{\infty}\frac{\sin nx}{n} = \frac{\pi - x}{2}

Across x=0x = 0 the value jumps from π2\dfrac{\pi}{2} to −π2-\dfrac{\pi}{2}, and that repeats every 2π2\pi. At the jump itself, however, the series converges to 00, the average of the two sides. Here is the fact that summing continuous functions can converge pointwise to something discontinuous.

What three terms look like

With only three terms the approach to the line is still crude. Even so, the skeleton of a sawtooth is already visible: the value falls broadly as xx runs from 00 to π\pi. Adding terms hugs the line more closely and steepens the rise at the jump.

The overshoot, or Gibbs phenomenon

A partial sum swells past the limit near a jump. With these three terms the largest excess appears at x=π4x = \dfrac{\pi}{4}.

QuantityValue
Maximum of the three-term sum at x=π4x = \dfrac{\pi}{4}≈1.4428\approx 1.4428
The limit π−x2\dfrac{\pi - x}{2} at the same point≈1.1781\approx 1.1781
Relative overshootabout 22%22\%

Adding terms does not remove that swelling; it narrows in width, its height staying at about 9%9\% of the jump. This is the Gibbs phenomenon, the sign that a Fourier series does not converge uniformly2. It is also the cause of the ringing seen around sharp changes in digital signal processing.

The derivative

The derivative is cos⁡x+cos⁡2x+cos⁡3x\cos x + \cos 2x + \cos 3x. Extended to infinitely many terms, ∑cos⁡nx\sum\cos nx does not converge in the ordinary sense and has meaning only as a distribution containing a Dirac delta. That corresponds to the fact that differentiating a sawtooth gives a constant slope together with an impulse at each jump.

Applications

A Fourier series is the tool for writing any periodic function as a superposition of sine waves.

  • The synthesis and analysis of sound
  • The compression of images
  • The solution of partial differential equations

The first step is exactly this question: what happens if one stops after finitely many terms.

  1. Fourier series, Wikipedia
  2. Gibbs phenomenon, Wikipedia