Fourier Series Calculator

Periodic decomposition

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About Fourier Series Calculator

A Fourier series calculator for periodic functions. Computes a₀/2 + Σ(aₙcos(nπx/L) + bₙsin(nπx/L)) coefficients using numerical integration. Shows coefficient table. Evaluate partial sums. Select from preset waveforms (square, sawtooth, triangle). All calculations are client-side.

Fourier Series Calculator Features

  • Coefficients
  • Partial sums
  • Waveform presets
  • n terms
  • Evaluation
Fourier series: f(x) = a₀/2 + Σₙ(aₙcos(nπx/L) + bₙsin(nπx/L)). aₙ = (1/L)∫f(x)cos(nπx/L)dx. bₙ = (1/L)∫f(x)sin(nπx/L)dx. Any periodic function can be decomposed into sines and cosines.

How to Use

Select a waveform:

  • Function: Square, sawtooth, ...
  • Terms: Number of harmonics
  • Evaluate: x value

Gibbs Phenomenon

At discontinuities, partial sums overshoot by ~9%. More terms don't fix it — the overshoot narrows but doesn't shrink. This is a fundamental property of Fourier convergence at jumps.

Applications

  • Signal processing (frequency analysis)
  • Heat equation solutions
  • Audio synthesis
  • Image compression

Step-by-Step Instructions

  1. 1Select a waveform.
  2. 2Set number of terms.
  3. 3View coefficients.
  4. 4Evaluate partial sum.
  5. 5Observe convergence.

Fourier Series Calculator — Frequently Asked Questions

Why do only odd harmonics appear in a square wave?+

Square waves have odd symmetry: f(x+L/2) = −f(x). This kills all even harmonics. The series is: (4/π)(sin(x) + sin(3x)/3 + sin(5x)/5 + ...). The 1/n decay is slow (not very smooth).

What is the Gibbs phenomenon?+

Near discontinuities, the Fourier partial sum overshoots by ~8.95% (integral of sinc). Adding more terms makes the overshoot narrower but not smaller. Named after Josiah Willard Gibbs (1899).

How fast do coefficients decay?+

Smoothness determines decay: continuous → 1/n², continuous derivative → 1/n³, etc. Discontinuous → 1/n (slowest). Analytic → exponential decay. Faster decay = faster convergence.

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