Fourier series

A Fourier series writes a function f(x) on an interval [0, L] as an infinite sum of sinusoids. For a function that vanishes at both endpoints, the expansion is f(x) = Σ bₙ sin(n π x / L), where the coefficients bₙ are integrals of f against each mode. Any reasonable shape — including ones with corners and sudden jumps — can be reconstructed this way.

Fourier series are the way physicists reach linearity. In a system where modes oscillate independently, writing a complicated initial shape as a sum of modes converts a hard PDE into a batch of easy ODEs, one per mode. This trick runs through the whole of wave physics, diffusion, quantum mechanics, optics, and signal processing. Its discrete cousin, the Fast Fourier Transform, is one of the most-executed algorithms in the world.

Joseph Fourier introduced the technique in his 1807 manuscript on heat flow and defended it at book length in 1822. Lagrange initially blocked its publication, arguing that discontinuous functions couldn't equal sums of continuous ones. The full rigorous theory took Dirichlet, Riemann, Lebesgue and most of the 19th century to establish.