Fourier Series Calculator

How to Use This Calculator

1. Select a Function Preset (Square Wave, Sawtooth, or Triangle).
2. Enter the Half-Period L (full period = 2L; default π gives period 2π).
3. Set the Number of Terms n (up to 20).
4. Enter x to evaluate the partial sum at that point.
5. The Professional tab shows complex coefficients |cₙ| and Parseval energy error.

Formula

Example

Frequently Asked Questions

• What is a Fourier series? ▼
  A Fourier series decomposes a periodic function into a sum of sine and cosine waves. Any periodic function f(x) with period 2L can be written as a₀ + Σ[aₙcos(nπx/L) + bₙsin(nπx/L)].
• What are the Fourier coefficients a₀, aₙ, bₙ? ▼
  a₀ = (1/2L)∫f(x)dx (DC offset); aₙ = (1/L)∫f(x)cos(nπx/L)dx; bₙ = (1/L)∫f(x)sin(nπx/L)dx — each integral over one full period.
• What is the Fourier series of a square wave? ▼
  The square wave has only odd harmonics: f(x) = (4/π)[sin(πx/L) + sin(3πx/L)/3 + sin(5πx/L)/5 + ...]. All cosine terms (aₙ) are zero due to odd symmetry.
• What is Parseval's identity? ▼
  Parseval's theorem states that the total energy in a signal equals the sum of squared Fourier coefficients: (1/L)∫|f|²dx = 2a₀² + Σ(aₙ²+bₙ²)/2. It connects signal energy to spectral energy.
• What is the complex exponential form of the Fourier series? ▼
  f(x) = Σcₙe^(inπx/L) where cₙ = (1/2L)∫f(x)e^(-inπx/L)dx. The magnitude |cₙ| gives the amplitude of the nth harmonic.

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