Fourier Series

Here is one of the most far-reaching ideas in all of mathematics: almost any function can be built out of sines and cosines. Feed in a jagged square wave, a saw-toothed ramp, the profile of a plucked string — and each is just a sum of smooth ripples of different frequencies, stacked in the right amounts.

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\Big(a_n\cos\frac{n\pi x}{L} + b_n\sin\frac{n\pi x}{L}\Big).

Joseph Fourier made the claim in 1822 while studying how heat flows, and his contemporaries flatly disbelieved it — how could ripples ever add up to a sharp corner? They can, and this course shows exactly how and why.

Why it matters

The trick that makes it all work is orthogonality: sines and cosines of different frequencies are "perpendicular" in the same precise sense that dot-product axes are perpendicular. That turns "find the right amounts" into a mechanical projection — one integral per coefficient.

Once you can take a function apart into frequencies, you can solve equations one frequency at a time. That is the engine behind the separation of variables method for the heat, wave, and Laplace equations — which is exactly why we build this toolkit first.

The path

  1. Orthogonal Functions — functions as vectors; why sines are perpendicular.
  2. The Fourier Coefficients — projecting onto each mode.
  3. The Fourier Series — assembling the full sum.
  4. Sine and Cosine Series — half-range expansions for boundary problems.
  5. Convergence & Gibbs — when the sum really equals the function.
  6. The Complex Fourier Series — the compact exponential form.
  7. The Fourier Transform — from a sum to an integral.

Begin → Orthogonal Functions