How do I find the Fourier series of a piecewise function?

Choose Custom / piecewise and enter each piece of one period with its start, end and formula, for example −1 from −π to 0 and 1 from 0 to π. The pieces must join end to end. The calculator integrates each piece separately and adds the results, which is exactly how you would split the integrals by hand.

What are a0, an and bn?

They are the Fourier coefficients. a₀/2 is the average value of the function over one period, aₙ is the weight of cos(nπx/L) and bₙ is the weight of sin(nπx/L). With period 2L: a₀ = (1/L)∫f dx, aₙ = (1/L)∫f cos(nπx/L) dx and bₙ = (1/L)∫f sin(nπx/L) dx over one period.

Why is the constant term a0/2 and not a0?

Defining a₀ with the same 1/L factor as aₙ makes one formula cover every n, so the average value is a₀/2. Some books define a₀ as the average directly; the constant term of the series is the same either way. This calculator shows a₀/2 as the DC term.

What is the Gibbs phenomenon?

At a jump discontinuity the partial sum overshoots the function by about 9% of the jump size, and the overshoot does not shrink as N grows — it only gets narrower. For a square wave from −1 to 1 the partial sums peak near 1.18. The calculator reports the peak of the partial sum so you can watch this.

How accurate are the numerical coefficients?

Simpson's rule with 4,000 steps per piece gives about six correct decimals for smooth pieces. Put any kink or jump at a piece boundary (for example split |x| into −x and x) for the best accuracy.

L is half the period, so the function repeats every 2L. L = π gives the textbook period 2π. For a custom function L is set by your pieces: from −1 to 1 gives L = 1.

Fourier Series Calculator

Enter any function — including a piecewise one such as 0 on (−π, 0) and x on (0, π) — or pick a standard waveform, choose N, and get a0, an, bn, the series written out and a graph of the partial sum.

Function Settings

Periodic Function

Period L = π (half-period)

12π452π

Harmonics N = 5

11020304050

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

Fourier Coefficients of Common Waveforms

Period 2L, series a₀/2 + Σ[aₙ cos(nπx/L) + bₙ sin(nπx/L)]. Odd functions have only sine terms; even functions only cosine terms.

Waveform	One period	a₀	aₙ	bₙ	Symmetry
Square wave	−1 on (−L, 0), 1 on (0, L)	0	0	4/(nπ), odd n only	Odd
Sawtooth	x/L on (−L, L)	0	0	2(−1)ⁿ⁺¹/(nπ)	Odd
Triangle	1 − 2\|x\|/L on (−L, L)	0	8/(n²π²), odd n only	0	Even
Full-wave rectified sine	\|sin(πx/L)\|	4/π	−4/(π(n²−1)), even n only	0	Even
Half-wave rectified sine	sin(πx/L) on (0, L), 0 on (L, 2L)	2/π	−2/(π(n²−1)), even n only	½ for n = 1, else 0	Neither
Parabola	x² on (−π, π)	2π²/3	4(−1)ⁿ/n²	0	Even

Quick answer

For a function with period 2L, the Fourier series is f(x) ≈ a₀/2 + Σ [aₙ cos(nπx/L) + bₙ sin(nπx/L)], with a₀ = (1/L)∫f(x)dx, aₙ = (1/L)∫f(x)cos(nπx/L)dx and bₙ = (1/L)∫f(x)sin(nπx/L)dx taken over one full period. Odd functions have every aₙ = 0 and even functions have every bₙ = 0. This calculator evaluates those integrals numerically for any function or set of pieces you type in.

Examples

f(x) on one period	a₀	aₙ	bₙ
Square wave −1 on (−π, 0), 1 on (0, π)	0	0	4/(nπ) for odd n, 0 for even n
x on (−π, π) (sawtooth)	0	0	2(−1)ⁿ⁺¹/n
x² on (−π, π)	2π²/3	4(−1)ⁿ/n²	0
\|sin x\| (full-wave rectified)	4/π	−4/(π(n²−1)) for even n	0

Choose Custom / piecewise to type your own function, or pick a waveform, then set the number of terms N.

Summary

This Fourier series calculator finds the coefficients a₀, aₙ and bₙ of a periodic function and shows how the partial sum with N terms approaches it. Type your own function of x, or split one period into up to six pieces (for example 0 on (−π, 0) and x on (0, π)), and the coefficients are computed by Simpson's-rule integration over one period. Standard waveforms — square, sawtooth, triangle, full-wave and half-wave rectified sine — use exact closed-form coefficients. The graph overlays the partial sum on the original function so you can see convergence and the Gibbs overshoot at jumps, and the table lists aₙ, bₙ and the amplitude of every harmonic up to N = 50.

How it works

Pick a standard waveform, or choose Custom / piecewise and enter one period as one or more pieces: start, end and f(x) for each. The period is the end of the last piece minus the start of the first, and L is half of it.
You can type x, numbers, + − × ÷ ^, pi (or π), e, and functions such as sin, cos, tan, exp, ln, log (base 10), sqrt and abs, for example 2x^2 − 1 or abs(sin(x)).
The calculator integrates a₀ = (1/L)∫f dx, aₙ = (1/L)∫f cos(nπx/L) dx and bₙ = (1/L)∫f sin(nπx/L) dx over the period with Simpson's rule (4,000 steps per piece). Standard waveforms use their exact formulas.
The partial sum S_N(x) = a₀/2 + Σ from n = 1 to N of [aₙ cos(nπx/L) + bₙ sin(nπx/L)] is drawn over two periods together with the original function.
The table lists aₙ, bₙ and the amplitude √(aₙ² + bₙ²) of each harmonic; the peak of the partial sum shows the Gibbs overshoot next to jumps.

Use cases

Check hand-calculated Fourier coefficients for a homework or exam problem, including piecewise functions.
See the Gibbs phenomenon: the roughly 9% overshoot that stays at a jump however many terms you add.
Find how many terms a square, sawtooth or triangle wave needs for a good approximation.
Get the harmonic amplitudes of a periodic signal for a signals and systems course.
Confirm symmetry shortcuts: odd functions have only sine terms and even functions only cosine terms.

Frequently Asked Questions

Last updated: 2026-10-10 · Reviewed by Nham Vu