ganak.appFourier Series Calculator
Free Fourier series calculator: get a₀, aₙ, bₙ coefficients, the partial-sum graph, and step-by-step derivation for any periodic function.
Coefficients
a₀ and 8 harmonics
| n | aₙ | bₙ |
|---|---|---|
| 0 | 0 | — |
| 1 | 0 | 1.999 |
| 2 | 0 | -0.9995 |
| 3 | 0 | 0.6663 |
| 4 | 0 | -0.4997 |
| 5 | 0 | 0.3998 |
| 6 | 0 | -0.3332 |
| 7 | 0 | 0.2856 |
| 8 | 0 | -0.2499 |
Step-by-Step Solution
How the coefficients are derived
Identify period & frequency
Interval [-3.14, 3.14], T = 6.28, ω = 2π/T = 1.0005.
Compute a₀ (average value)
a₀ = (1/T) ∫ f(x) dx = 0.
Compute aₙ (cosine coefficients)
aₙ = (2/T) ∫ f(x) cos(nωx) dx for n = 1 .. 8.
Compute bₙ (sine coefficients)
bₙ = (2/T) ∫ f(x) sin(nωx) dx for n = 1 .. 8.
Assemble the series
f(x) ≈ a₀ + Σ [aₙ cos(nωx) + bₙ sin(nωx)], n = 1 .. 8.
What Is a Fourier Series?
Decomposing any periodic function into sines and cosines
A Fourier series rewrites a periodic function as a sum of sines and cosines. Square waves, triangle waves, sawtooth ramps, and even discontinuous step functions can all be approximated by adding enough harmonics together.
Core Idea
f(x) ≈ a₀ + Σ [aₙ cos(nωx) + bₙ sin(nωx)]
f(x)
Periodic function
a₀
DC / average term
aₙ, bₙ
Harmonic amplitudes
ω = 2π/T
Angular frequency
The technique is essential in signal processing (decomposing a waveform into frequency components), PDEs (separation of variables for the heat and wave equations), and data compression (JPEG, MP3, MPEG all rely on Fourier-style basis representations).
Coefficient Formulas by Series Type
Five modes, three integral families
1. Full Fourier Series
Both sines and cosines on a symmetric interval. The default mode for general periodic functions on [−L, L].
a₀ = (1/T) ∫ₐᵇ f(x) dx
aₙ = (2/T) ∫ₐᵇ f(x) cos(nωx) dx
bₙ = (2/T) ∫ₐᵇ f(x) sin(nωx) dx
2. Half-Range Sine Series
Odd extension on [0, L]. Vanishes at both endpoints — used when boundary conditions are f(0) = f(L) = 0.
f(x) ≈ Σ bₙ sin(nπx/L) · bₙ = (2/L) ∫₀ᴸ f(x) sin(nπx/L) dx
3. Half-Range Cosine Series
Even extension on [0, L]. Zero slope at endpoints — used when boundary conditions are f′(0) = f′(L) = 0.
f(x) ≈ a₀ + Σ aₙ cos(nπx/L) · aₙ = (2/L) ∫₀ᴸ f(x) cos(nπx/L) dx
4. Complex (Exponential) Series
Algebraically cleaner. Used in signal processing and as the bridge to the Fourier transform.
f(x) ≈ Σ cₙ e^(inωx) · cₙ = (1/T) ∫ₐᵇ f(x) e^(−inωx) dx
Quick Reference & Worked Example
Standard textbook results at a glance
| Function on [−π, π] | Non-zero coefficients | Parity |
|---|---|---|
| x | bₙ = (−1)^(n+1) · 2/n | Odd |
| x² | a₀ = π²/3, aₙ = 4(−1)^n/n² | Even |
| |x| | a₀ = π/2, aₙ = −4/(πn²) (odd n) | Even |
| Square wave (±1) | bₙ = 4/(nπ) (odd n) | Odd |
| Triangle wave | aₙ = −4/(π·n²) (odd n) | Even |
| Sawtooth (= x) | bₙ = (−1)^(n+1) · 2/n | Odd |
Worked Example: Square Wave
Decompose f(x) = −1 on (−π, 0) and +1 on (0, π):
Period T = 2π ω = 1
Step 1: f(x) is odd → a₀ = 0 and every aₙ = 0.
Step 2: bₙ = (2/2π) · ∫₋π^π f(x) sin(nx) dx = (2/π) · ∫₀^π sin(nx) dx
Step 3: Evaluate: bₙ = (2/π) · [(1 − cos(nπ))/n] = (2/π) · (1 − (−1)^n)/n
Step 4: bₙ = 4/(nπ) for odd n; bₙ = 0 for even n.
Result: f(x) = (4/π)[sin(x) + sin(3x)/3 + sin(5x)/5 + …] ✓
Where Fourier Series Are Used
Applications across science and engineering
Audio & Signal Processing
Decomposes a waveform into frequency components for filtering, equalization, and lossy compression in MP3/AAC.
Heat & Wave Equations
Separation of variables in PDEs naturally produces Fourier expansions; coefficients fix the boundary/initial data.
Image Compression (JPEG)
JPEG uses a discrete cosine transform — a half-range Fourier cosine series on 8×8 image blocks.
Communications
Modulation, demodulation, and OFDM in 4G/5G/Wi-Fi all rely on Fourier representations of signals.
Quantum Mechanics
Solutions to the Schrödinger equation in a periodic potential expand in plane-wave (Fourier) basis.
Electrical Engineering
Steady-state response of linear circuits to non-sinusoidal periodic inputs uses harmonic-by-harmonic analysis.
Common Mistakes & Convergence
The pitfalls that show up in homework problems
Wrong period
Set the interval to one full period of the periodic extension. For a square wave centred at 0, that is [−π, π], not [0, π].
Forgetting the (2/T) factor
aₙ and bₙ have a 2/T scaling; a₀ has only 1/T. Mixing them produces coefficients off by a factor of 2.
Ignoring even/odd parity
If f(x) is even, every bₙ vanishes. If it is odd, every aₙ (and a₀) vanishes. Use a half-range mode to bake this in.
Gibbs overshoot near jumps
At a jump discontinuity the partial sum overshoots by ~9% of the jump height regardless of N. The overshoot narrows as N grows; it does not disappear.
Related Calculators
Tools that compose well with Fourier analysis
Frequently Asked Questions
Common questions about Fourier series and how to compute them
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Last updated Apr 28, 2026