Fourier Series Calculator

A Fourier series represents a periodic function as an infinite sum of sine and cosine terms (harmonics). Any function that satisfies the Dirichlet conditions (finite number of discontinuities and maxima/minima per period) can be expressed as f(x) = a0/2 + sum of (an*cos(n*x) + bn*sin(n*x)) for n from 1 to infinity.

For a 2*pi-periodic function f(x): a0 = (1/pi)*integral of f(x) dx from -pi to pi; an = (1/pi)*integral of f(x)*cos(n*x) dx; bn = (1/pi)*integral of f(x)*sin(n*x) dx. For odd functions (like square wave and sawtooth), all cosine coefficients an are zero. For even functions, all sine coefficients bn are zero.

Near jump discontinuities in a periodic function, the Fourier series partial sum overshoots the true value by about 9% of the jump height, regardless of how many terms are included. This overshoot does not decrease with more terms; instead, it becomes narrower. This is called the Gibbs phenomenon, named after J. Willard Gibbs.

A Fourier series applies to periodic functions and decomposes them into discrete frequency components (harmonics at n*omega0 for integer n). The Fourier transform applies to non-periodic functions and produces a continuous spectrum of frequencies. The discrete Fourier transform (DFT) used in signal processing is a related but distinct operation.

Fourier series are used in signal processing (audio, radio), image compression (JPEG uses the discrete cosine transform, a variant), solving partial differential equations (heat equation, wave equation), analysing vibrations and acoustics, and electrical engineering for analysing periodic signals and power systems.