What is a Fourier series?

A Fourier series decomposes a periodic function into an infinite sum of sine and cosine waves. The coefficients a0, an, and bn measure how strongly each frequency contributes to the original waveform.

What does the a0 term represent?

a0/2 is the DC offset — the average value of the function over one period. For symmetric functions like the square and triangle wave, this is zero.

Why does the square wave overshoot near edges?

That is the Gibbs phenomenon. No matter how many terms you add, the partial sum overshoots by about 9% at a jump discontinuity. It is a fundamental property of Fourier series near discontinuities.

What is the difference between an and bn coefficients?

an multiplies cos(nπx/L) — it captures even symmetry in the function. bn multiplies sin(nπx/L) — it captures odd symmetry. A purely odd function (like the sawtooth) has all an = 0.

Can I use a custom function?

Currently the tool provides four built-in waveforms with exact closed-form coefficients. Custom function support would require numerical integration, which may be added in a future update.

What is the period L?

L is the half-period — the function repeats every 2L. Setting L = π means one full period spans 2π, which matches the standard textbook convention.

Fourier Series Calculator

Name: Fourier Series Calculator
Availability: InStock
Author: Nham Vu

Select a periodic function, set the number of terms, and instantly see the Fourier coefficients plus a live graph of the partial sum approximation.

Function Settings

Periodic Function

Period L = π (half-period)

12π452π

Harmonics N = 5

1102030

Show original function

Fourier Series Formula

DC Term (a₀/2)

Average value of f(x) over one period.

Partial Sum Approximation

Fourier sum Original

Coefficient Table

n	a_n	b_n	Amplitude

Summary