Piecewise Function Fourier Series Calculator

Analyze periodic functions across intervals with flexible harmonics. Review coefficients, approximations, symmetry, and sampled values. Download outputs, compare graphs, and study convergence with confidence.

Results

Series Summary

Series Expression

Function and Fourier Approximation

Coefficient Magnitudes

Fourier Coefficients

n	a_n	b_n	\|c_n\| approx

Sample Comparison

x	f(x)	F_N(x)	Error

Calculator

Half-period L

The series is computed on [-L, L].

Number of harmonics N

Number of pieces

Integration steps

Plot periods

Plot samples

Piecewise Definition

Use expressions in x. Interval entries may use values like -pi, 0, pi/2, or 3.

Example Data Table

This sample shows a simple piecewise function on [-π, π].

Piece	Interval	Definition	Meaning
1	[-π, 0)	f(x) = x	Linear negative ramp
2	[0, π]	f(x) = π - x	Descending positive segment

Formula Used

For a periodic function defined on [-L, L], the real Fourier series is:

f(x) ≈ a₀/2 + Σ [ a_n cos(nπx/L) + b_n sin(nπx/L) ]

The coefficients are computed with numerical integration:

a₀ = (1/L) ∫_-L^L f(x) dx
a_n = (1/L) ∫_-L^L f(x) cos(nπx/L) dx
b_n = (1/L) ∫_-L^L f(x) sin(nπx/L) dx

This calculator evaluates the piecewise function over many sample points, applies the trapezoidal rule, and builds the N-term partial sum. It also reports approximation error and coefficient magnitudes.

How to Use This Calculator

Enter the half-period L for your periodic interval.
Choose the number of harmonics N.
Set how many function pieces you need.
Enter each interval and expression in x.
Adjust integration and plotting settings if required.
Click Compute Fourier Series.
Review coefficients, graphs, and sample error values.
Export the output as CSV or PDF when needed.

FAQs

1. What does this calculator compute?

It computes the real Fourier series of a piecewise function over one period. The tool estimates a0, an, bn, the partial sum, error values, and visual graphs.

2. Can I enter trigonometric or exponential expressions?

Yes. You can enter expressions such as sin(x), cos(2*x), exp(x), x^2, abs(x), or combinations supported by the parser.

3. Do the intervals need to cover the whole domain?

No, but uncovered points are treated as zero. For accurate modeling, define pieces so they represent the full interval from -L to L.

4. Why do jumps create oscillations near boundaries?

That behavior is normal. Finite Fourier sums near discontinuities show Gibbs oscillation, which shrinks in width as harmonics increase.

5. What happens if my function is even or odd?

Even functions tend to produce very small sine coefficients. Odd functions tend to produce very small cosine coefficients and a0 close to zero.

6. Are the coefficients exact?

No. They are numerical approximations based on the trapezoidal rule. Increasing integration steps usually improves accuracy.

7. Why should I increase the number of harmonics?

More harmonics usually improve the partial sum and capture sharper features. The trade-off is longer computation and denser output tables.

8. What is included in the downloads?

The CSV and PDF exports include summary values, Fourier coefficients, and representative sampled comparisons from the current calculation.

Notes

This page uses a numerical approach for general piecewise inputs. It is suitable for classroom work, engineering approximations, and quick validation of periodic models.