Fourier series

Any reasonable periodic function is a sum of sines and cosines of multiples of a base frequency: $f(t) = \sum_n c_n e^{2\pi i n t/T}$. The coefficients $c_n$ — the spectrum — say how much of each harmonic it contains.

Fourier (1807) needed to solve the heat equation: sines are the shapes that keep their form while decaying, so if any initial temperature can be written as a sum of sines, the problem splits into independent easy pieces. The claim that any function could be so written was shocking — and it changed analysis.

Sines of different frequencies are orthogonal, like perpendicular axes. The coefficient $c_n$ is the projection of $f$ onto the $n$-th harmonic, $\frac1T\int f(t)e^{-2\pi int/T}\,\dd t$ — a correlation that measures how much $f$ "resonates" at that frequency. A square wave is the sum of odd harmonics $\frac{4}{\pi}\big(\sin t + \frac13\sin 3t + \frac15\sin5t + \dots\big)$; its jump causes a 9% overshoot that never disappears (Gibbs phenomenon) — the ringing you see around edges in over-compressed JPEGs.

For $f$ with period $T$ and $\int_0^T|f|^2 < \infty$,

with convergence in mean square, and pointwise wherever $f$ is smooth. Parseval: $\frac1T\int_0^T|f|^2 = \sum|c_n|^2$ (energy is the same in time and frequency).

Musical timbre is the pattern of harmonic amplitudes; audio equalizers boost or cut frequency bands; the heat and wave equations are solved term by term; and the discrete version (DFT/FFT) is the workhorse of digital signal processing.