Calculator Form
Example Data Table
This example uses f(x) = x² on [0, π]. You can paste these pairs into manual mode.
| x | f(x) = x² |
|---|---|
| 0.000000 | 0.000000 |
| 0.523599 | 0.274156 |
| 1.047198 | 1.096623 |
| 1.570796 | 2.467401 |
| 2.094395 | 4.386490 |
| 2.617994 | 6.853892 |
| 3.141593 | 9.869604 |
Formula Used
This calculator estimates the integrals with the trapezoidal rule, which works well for dense smooth samples and remains practical for manual datasets.
How to Use This Calculator
- Choose a built-in function or switch to manual x,y pairs.
- Set the interval length L and the number of cosine terms.
- Adjust sampling density, reconstruction points, and the evaluation x value.
- Submit the form to compute coefficients, errors, and the reconstructed series.
- Review the graphs, inspect the tables, then export CSV or PDF files.
Frequently Asked Questions
1) What does this calculator compute?
It estimates half-range cosine Fourier coefficients from sampled data or built-in functions, reconstructs the signal, and reports approximation error over the interval [0, L].
2) When should I use a cosine series?
Use it when the function is defined on [0, L] and an even extension is suitable. This appears often in boundary-value problems and symmetric physical models.
3) Does adding more terms always help?
More terms usually improve overall accuracy for smooth signals. Near jumps, oscillations can remain because of the Gibbs effect, even when many harmonics are included.
4) Why is trapezoidal integration used?
It is simple, stable, and effective for sampled data. With enough points, it gives reliable coefficient estimates without requiring symbolic integration.
5) Can I enter irregularly spaced points?
Yes. Manual mode accepts nonuniform x values as long as they stay within [0, L]. The series coefficients are still integrated numerically from those samples.
6) What does the evaluation point show?
It compares the original function value and the reconstructed series value at one chosen x location. This helps you inspect local approximation quality.
7) Why might errors stay large near sharp corners?
Corners and discontinuities need many harmonics. The approximation converges globally, but localized ringing or slow pointwise improvement can still appear near abrupt changes.
8) What should I export?
Use the coefficient CSV for numerical analysis, the points CSV for plotting elsewhere, and the PDF when you need a clean summary for reports or coursework.