Fourier series

A Fourier series rests on one structural fact: sines and cosines of integer multiples of a base frequency are mutually orthogonal over one period, in exactly the sense that lets you pick out "how much" of each one is present in a target function the same way a dot product picks out a vector's component along an axis. Multiplying the function by (\cos(n\omega t)) or (\sin(n\omega t)) and integrating over one period isolates that one coefficient, because every other term in the sum integrates to zero against it.

This only works, and the resulting series only converges, when the function satisfies the Dirichlet conditions: finitely many discontinuities and finitely many maxima/minima per period, and an absolutely integrable function. Most functions an engineer or physicist actually writes down satisfy this comfortably, but "any periodic function at all" does not.

Convergence itself has a subtlety worth remembering: at a genuine jump discontinuity, the series does not converge to either side's value — it converges to the average of the left- and right-hand limits, overshooting slightly near the jump (the Gibbs phenomenon). And a symmetry shortcut is worth knowing before grinding through integrals: an even function's series contains only cosines, an odd function's only sines, because the orthogonality integral for the "wrong" symmetry vanishes identically.