Fourier Series Calculator

What Is a Fourier Series? A Fourier series represents a periodic function as an infinite sum of sine and cosine waves: f(x) = a₀/2 + Σ[aₙcos(nπx/L) + bₙsin(nπx/L)] The idea is that ANY periodic function, no matter how irregular, can be built from pure sinusoids. Jean-Baptiste Joseph Fourier developed this in 1822 in France to solve the heat equation. His claim was initially controversial — Lagrange and Laplace doubted it — but it was ultimately proven correct.

Fourier Coefficients a₀ = (1/L) ∫₋L^L f(x) dx (twice the average value) aₙ = (1/L) ∫₋L^L f(x)cos(nπx/L) dx (cosine coefficients) bₙ = (1/L) ∫₋L^L f(x)sin(nπx/L) dx (sine coefficients) For even functions (f(−x) = f(x)): only aₙ coefficients (cosine series). For odd functions (f(−x) = −f(x)): only bₙ coefficients (sine series).

Common Fourier Series Square wave (period 2π, amplitude 1): f(x) = (4/π) × [sin(x) + sin(3x)/3 + sin(5x)/5 + …] = (4/π) × Σ sin((2k−1)x)/(2k−1) Only odd harmonics appear. Coefficients decay as 1/n.

Sawtooth wave (period 2π): f(x) = 2 × [sin(x) − sin(2x)/2 + sin(3x)/3 − …] = 2 × Σ (−1)^(n+1) sin(nx)/n All harmonics present. Coefficients decay as 1/n.

Triangle wave (period 2π): f(x) = (8/π²) × [sin(x) − sin(3x)/9 + sin(5x)/25 − …] = (8/π²) × Σ (−1)^k sin((2k+1)x)/(2k+1)² Only odd harmonics. Coefficients decay as 1/n².

Gibbs Phenomenon Near a jump discontinuity, the Fourier partial sum overshoots by about 9% of the jump height. This overshoot does NOT disappear as more terms are added — it narrows but stays at ~9%. Named after J. Willard Gibbs (1899 in the United States), though first noticed by Wilbraham in 1848. Gibbs phenomenon is a fundamental limitation of Fourier series at discontinuities.

Parseval’s Theorem The sum of squared coefficients equals the mean squared value of the function: (a₀/2)² + Σ(aₙ² + bₙ²)/2 = (1/2L) ∫₋L^L [f(x)]² dx This is an energy conservation law in frequency space — total power = sum of power in each harmonic. Famous use: Σ 1/n² = π²/6 (Basel problem) follows from Parseval applied to the sawtooth wave.

Applications Signal processing: any periodic signal decomposes into harmonics — the basis of spectrum analyzers. Audio: musical tones have fundamental + harmonics. A violin and a flute playing the same note differ by their harmonic content (timbre). Image compression: JPEG uses the discrete cosine transform (DCT), a close relative of the Fourier series. Heat equation: temperature distribution in a rod is a Fourier series in spatial coordinate.