An Etymological Dictionary of Astronomy and Astrophysics

A theorem relating the → Fourier coefficients to the function that they describe. It states that:

(1/L) ∫ |f(x)|²dx (integrated from x₀ to x₀ + L) = (a₀/2)² + (1/2) Σ (a_r² + b_r²) (summed from r = 1 to ∞).

In other words, the sum of the moduli squared of the complex Fourier coefficients is equal to the average value of |f(x)|² over one period.

See also: Named after Marc-Antoine Parseval (1755-1836), French mathematician; → theorem.