Calculator Input
Enter a function in x. Use explicit multiplication like 2*x. Supported functions include sin, cos, tan, exp, log, sqrt, and abs.
Example Data Table
Use these sample setups to test the calculator quickly.
| Function | Interval | Harmonics | Segments | Observation |
|---|---|---|---|---|
| x | [-π, π] | 8 | 1600 | Produces a mainly sine-based odd series. |
| x^2 | [-π, π] | 10 | 2000 | Produces an even series with cosine terms. |
| abs(x) | [-π, π] | 12 | 2200 | Shows strong cosine harmonics and corner effects. |
| sin(x)+0.5*cos(2*x) | [-π, π] | 6 | 1400 | Matches the original blend with few terms. |
Formula Used
The Fourier series on the interval [-L, L] is written as a constant term plus cosine and sine harmonics.
Series form:
f(x) ≈ a0/2 + Σ [ an cos(nπx/L) + bn sin(nπx/L) ]
Coefficient formulas:
a0 = (1/L) ∫-LL f(x) dx
an = (1/L) ∫-LL f(x) cos(nπx/L) dx
bn = (1/L) ∫-LL f(x) sin(nπx/L) dx
This page estimates the integrals numerically using midpoint integration. Higher segment counts improve accuracy. Higher harmonic counts improve the series fit, especially for smooth functions.
How to Use This Calculator
- Choose a preset or enter your custom function.
- Set the half-period L for the interval [-L, L].
- Select the number of harmonics N.
- Set numerical segments for coefficient integration.
- Enter x₀ if you want a point evaluation.
- Press Find Fourier Series to generate results.
- Review the coefficient table and Plotly graphs.
- Download the report as CSV or PDF.
Frequently Asked Questions
1) What functions can I enter?
You can enter expressions in x using operators and common functions. Examples include x, x^2, abs(x), sin(3*x), exp(x), and sqrt(abs(x)+1).
2) What does L mean here?
L is the half-period of the interval. The calculator builds the Fourier series over [-L, L]. The full period is therefore 2L.
3) Why do I see sine and cosine terms together?
General functions usually need both term types. Even functions mainly produce cosine terms. Odd functions mainly produce sine terms. Mixed symmetry usually creates both.
4) Why is the result approximate?
This tool uses numerical integration and a finite number of harmonics. That makes the result a practical approximation instead of a symbolic closed-form derivation.
5) How many harmonics should I use?
Smooth functions often look good with fewer terms. Sharp corners or jumps need more harmonics. Start with 8 to 12 and increase gradually.
6) Why do oscillations appear near jumps?
That is the Gibbs effect. Truncated Fourier series can overshoot near discontinuities. More harmonics narrow the region, but the local overshoot does not fully disappear.
7) What improves accuracy most?
Increase numerical segments for better integral estimates. Increase harmonics for better curve fitting. Both settings work together to improve the final approximation.
8) Can I export my calculated coefficients?
Yes. Use the CSV button for spreadsheet-friendly data. Use the PDF button for a formatted summary containing series details and coefficient tables.