trigonometric Fourier series
/ FOOR-ee-ay /
Pluck a guitar string and a single note carries a whole stack of overtones; speak into a microphone and the wiggly voltage trace is really many pure tones added together. The trigonometric Fourier series is the mathematical version of that idea: it claims that almost any repeating signal, no matter how jagged, can be rebuilt by adding up plain sines and cosines whose frequencies are whole-number multiples of one fundamental rate. It is the founding move of harmonic analysis — the discovery that a complicated periodic shape is secretly a chord.
Precisely, if f(x) repeats with period 2L, its trigonometric Fourier series is f(x) = a_0/2 + sum over n from 1 to infinity of [ a_n cos(n pi x / L) + b_n sin(n pi x / L) ]. The constant term a_0/2 is the average (DC) level of the signal; each a_n and b_n says how much of the n-th cosine and sine you must mix in. The single number n indexes the harmonics: n = 1 is the fundamental, n = 2 the first overtone, and so on. The pieces cos(n pi x / L) and sin(n pi x / L) all share the period 2L, so their sum is automatically 2L-periodic, matching f. Building the partial sum up to some N gives a finite trigonometric polynomial that approximates f, getting better as N grows wherever f is smooth.
This expansion is the gateway to the methods of mathematical physics. Because sines and cosines are the natural shapes of vibration and the eigenfunctions of the second-derivative operator on an interval, writing a signal as a Fourier series turns problems about heat flow, wave motion, and electrical signals into bookkeeping on a list of coefficients — each harmonic evolves independently. The catch worth remembering up front: equality 'f(x) = the series' holds in a careful, mostly-everywhere sense, and at corners or jumps the series does something special rather than tracking f exactly there.
The square wave equal to +1 on (0, pi) and -1 on (-pi, 0), extended with period 2pi, has Fourier series (4/pi) [ sin(x) + sin(3x)/3 + sin(5x)/5 + ... ] — only odd sines, with coefficients dying off like 1/n.
An infinitely sharp square wave reassembled from smooth sine curves — the textbook first example, and the source of the famous overshoot at the jumps.
A Fourier series needs a period: it represents f only on one period and then repeats that picture forever, so applying it to a non-periodic f silently periodizes it and may introduce jumps at the interval ends that were never in the original.