Fourier's Series

[guitar intro] "Fourier's theorem" redirects here. For the number of real roots of a polynomial, see Budan's theorem § Fourier's theorem. Fourier transforms Fourier transform Fourier series Discrete-time Fourier transform Discrete Fourier transform Discrete Fourier transform over a ring Fourier transform on finite groups Fourier analysis Related transforms A Fourier series (/ˈfʊrieɪ, -iər/[1]) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series.[2] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. [guitar solo] (Fourier series) [guitar solo] (fourier series) [epic orchestral outro]