Parseval's theorem
/ par-suh-VAL /
If a signal is the sum of its harmonics, where does its total energy go? Intuition says: it must be shared out among the harmonics, with no energy created or lost in the bookkeeping. Parseval's theorem makes that intuition exact. It says the average squared size of a function over one period equals the sum of the squared sizes of all its Fourier coefficients — the total 'energy' in the time picture equals the total 'energy' in the frequency picture.
In the trigonometric form, for f of period 2L, (1/L) integral over one period of [f(x)]^2 dx = (a_0)^2/2 + sum over n from 1 to infinity of (a_n^2 + b_n^2). In the complex form it is even cleaner: (1/(2L)) integral over one period of |f(x)|^2 dx = sum over all n of |c_n|^2. Either way the message is the same — squaring and averaging the signal gives the same number as squaring and adding up its spectral amplitudes. This is the energy interpretation: each harmonic contributes its own slice of the total, and the slices simply add, because the harmonics are orthogonal (no cross terms survive the integral).
Parseval is the workhorse identity behind power spectra and signal energy budgets. It lets you compute a mean-square value (RMS power, signal variance, the L^2 norm) directly from coefficients without ever reconstructing the signal, and it tells you how the energy is distributed across frequencies — the basis of every equalizer display and bandwidth calculation. As a bonus, applied to a known series it spits out striking number-theoretic sums; for instance the sawtooth's coefficients feed Parseval's identity to give sum of 1/n^2 = pi^2/6, the Basel sum.
For f(x) = x on (-pi, pi) with b_n = 2(-1)^{n+1}/n: Parseval gives (1/pi) integral from -pi to pi of x^2 dx = sum b_n^2, i.e. 2pi^2/3 = sum of 4/n^2, which rearranges to sum 1/n^2 = pi^2/6.
Energy conservation in the spectrum hands you a famous infinite sum as a free side effect.
Parseval requires the function to be square-integrable over the period; it measures total energy and is blind to phase and to pointwise behavior, so two functions differing only at isolated points (or in sign pattern) can share the same energy total.