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Concept

Eigenfunction

A function that an operation such as differentiation merely rescales, as a matrix rescales an eigenvector. Sines are eigenfunctions of the second derivative, which is why a vibrating string or a cooling rod is solved by breaking its shape into sine waves.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as heat equation, irreversibility — the same set of essays touches all of them, so they are one junction rather than several.

A square profile of heat, spreading. The same profile at four times, each drawn from the same harmonics with each one damped by the exponential of minus its frequency squared times the time. The corners go first.

The corners go first

Fourier was not decomposing waves for the pleasure of it. He was solving the flow of heat, and the whole apparatus exists because each harmonic fades at a rate set by the square of its frequency — which is why a sharp profile smooths instantly and why the flow cannot be run backwards.

analysis · Fourier series A plucked string, released. A string plucked into a tent shape at 0.5, drawn at times 0, 0.1, 0.25, 0.5, 0.75, 1: the corner splits into two corners that run apart, reflect off the ends upside down and meet again.

A plucked string keeps its corners

Heat smooths a sharp profile at once, because each harmonic decays at a rate set by the square of its frequency. Change one time derivative into two and nothing decays at all: each harmonic swings for ever, the corner of a pluck splits in two and runs along the string, and after one period the shape comes back exactly. The same sines, the same coefficients — and a flow that loses nothing.

analysis · Fourier series

Named alongside it

The objects these essays reach for when they reach for this one.

Fourier analysisHeat equationIrreversibilitySpectrumDiffusionExponential decayHarmonicsPeriodicitySuperpositionWave equation

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