Enter intervals, formulas, period, and harmonic depth precisely. Inspect tables, graphs, exports, and convergence details. Built for careful learning, testing, comparison, and repeated verification.
| Piece | Interval | Formula | Meaning |
|---|---|---|---|
| 1 | [0, π) | x | Rising linear segment |
| 2 | [π, 2π] | 2*pi - x | Falling linear segment |
| Period | [0, 2π] | T = 2π | Triangular wave over one full cycle |
This sample creates a triangular wave. It is useful for checking coefficient decay, graph shape, and how the partial sum improves as harmonics increase.
Let the chosen interval be [a, b]. The period is T = b - a and the fundamental angular frequency is ω₀ = 2π / T.
The real Fourier series is:
f(x) ≈ a₀/2 + Σ[aₙ cos(nω₀(x-a)) + bₙ sin(nω₀(x-a))]
The coefficients are computed numerically:
a₀ = (2/T) ∫ f(x) dx
aₙ = (2/T) ∫ f(x) cos(nω₀(x-a)) dx
bₙ = (2/T) ∫ f(x) sin(nω₀(x-a)) dx
This file uses the trapezoidal rule over many sample points. That makes the calculator flexible for piecewise formulas that may not have quick symbolic antiderivatives.
Enter the main interval first. This interval is treated as one period of the function.
Add each piece with a start, an end, and a formula in x. Make sure the listed pieces cover the whole interval.
Choose the harmonic depth N. Larger values usually improve detail but can reveal oscillation near jump points.
Set integration steps for coefficient accuracy and graph points for a smoother plot.
Press the calculate button. The page shows the result above the form, including coefficients, reconstruction samples, download options, and a graph.
Small higher-order amplitudes usually mean the function is smooth. Slower decay often means corners or jumps are present.
Near discontinuities, the partial sum may overshoot. That is a normal Gibbs-type effect, not always a calculation error.
Compare RMSE and maximum absolute error when you test different harmonic counts. Those values help you judge approximation quality quickly.
Enter one full period of the function. The calculator repeats that interval periodically when it reconstructs the Fourier series.
Yes. This file provides four piece rows. Use as many as needed, and leave unused rows empty.
The parser expects 2*x rather than 2x. Explicit multiplication reduces ambiguity and keeps expression evaluation predictable.
That is common in Fourier approximations of discontinuous functions. The oscillation narrows with more terms, but the local overshoot does not vanish completely.
a0 controls the constant part of the series. The average value of the function over one period is a0 divided by two.
Increase them when formulas change quickly, contain sharp bends, or when coefficient values seem unstable between runs.
They summarize each harmonic in compact form. Amplitude measures strength, while phase shows the horizontal shift of that harmonic component.
It is mainly a numerical tool. It is excellent for checking work, exploring shapes, and validating expected coefficient patterns.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.