Fourier series
Imagine a musical chord. The ear hears one rich sound, but a physicist would say it is built from pure tones — a fundamental pitch and its overtones, each at a definite loudness. A Fourier series does exactly this for a periodic function: it writes the function as a sum of pure sinusoidal tones, the sines and cosines whose frequencies are whole-number multiples of the basic period.
Concretely, for a function f that is integrable on an interval of length 2*pi and repeats with period 2*pi, the Fourier series is a_0/2 + sum over n >= 1 of (a_n cos(n x) + b_n sin(n x)). The numbers a_n and b_n are the Fourier coefficients, computed by integrating f against cos(n x) and sin(n x). One can equivalently use the complex form sum over all integers n of c_n e^{i n x}, which is tidier for analysis.
A central caveat: writing down the series is one thing; knowing it equals f is another. The series may converge pointwise to f, or only in the mean (L2), or — for a continuous function — it can even diverge at some points (du Bois-Reymond, 1873). So the symbol meaning ‘f has the series …’ is a statement about coefficients, not an automatic equality. Which mode of convergence holds, and under what hypotheses, is the heart of the subject.
The square wave f(x) = +1 on (0, pi) and -1 on (-pi, 0) has Fourier series (4/pi)(sin x + (sin 3x)/3 + (sin 5x)/5 + …). Only odd sine terms survive, because f is odd and has half-wave symmetry.
At the jump points x = 0, pi the series converges to 0, the average of the two one-sided values — a hint of Dirichlet's theorem.
The constant term is written a_0/2 (not a_0) precisely so that the single formula a_n = (1/pi) integral of f(x) cos(n x) dx works for every n including n = 0.