Calculator Inputs
Formula Used
For an even function on [-L, L], the sine terms vanish. The approximation becomes:
f(x) ≈ a₀/2 + Σ aₙ cos(nπx/L) for n = 1, 2, 3, ...
a₀ = (2/L) ∫₀ᴸ f(x) dxaₙ = (2/L) ∫₀ᴸ f(x) cos(nπx/L) dx
This calculator uses Simpson integration on [0, L]. It then reconstructs the truncated cosine series using the chosen number of harmonics.
How to Use This Calculator
- Enter the half-range L for the symmetric interval [-L, L].
- Select how many harmonics you want in the cosine series.
- Choose the integration step count for coefficient accuracy.
- Pick an even function type, then fill its parameters.
- Enter x₀ if you want a pointwise comparison.
- Press Calculate Fourier Series to generate coefficients, summary values, and graphs.
- Use the CSV or PDF buttons to export the computed coefficient table.
Example Data Table
Sample case: choose an even polynomial with L = π, c₀ = 0, c₂ = 1, c₄ = 0, c₆ = 0. This means f(x) = x² on [-π, π].
| Input Item | Value | Meaning |
|---|---|---|
| Function Type | Even Polynomial | Creates x² when c₂ = 1 and other coefficients are zero. |
| L | 3.141593 | Half-range for the interval [-π, π]. |
| Harmonics | 3 | Uses the first three cosine coefficients. |
| a₀ | 6.579736 | So the constant series part is a₀/2 = π²/3. |
| a₁ | -4.000000 | First cosine coefficient for x². |
| a₂ | 1.000000 | Second cosine coefficient for x². |
| a₃ | -0.444444 | Third cosine coefficient for x². |
FAQs
1) What does this calculator actually compute?
It computes the cosine coefficients of an even function on [-L, L], builds a truncated Fourier series, compares the approximation with the original function, and reports error measures.
2) Why are there no sine coefficients here?
Even functions satisfy f(-x) = f(x). Their Fourier sine terms cancel by symmetry, so only cosine terms remain in the final series.
3) Why does increasing harmonics improve the graph?
More harmonics add finer cosine detail. This usually reduces truncation error and makes the reconstructed curve follow the original function more closely.
4) What do integration steps control?
They control numerical accuracy when estimating the integrals for each coefficient. More steps usually improve stability, especially for sharp pulses or highly curved functions.
5) Can I use x² or other polynomial examples?
Yes. Select Even Polynomial and assign coefficients to c₀, c₂, c₄, and c₆. For x², use c₂ = 1 and set the others to zero.
6) Why can the approximation oscillate near jumps?
Truncated Fourier series often show edge oscillations near discontinuities. This is a normal Gibbs-type effect and does not mean the coefficient formulas are wrong.
7) What does the point error at x₀ show?
It shows the difference between the truncated cosine series and the original function at your selected evaluation point. Smaller values indicate a closer local fit.
8) What is the best function type for testing convergence?
Smooth functions such as gaussians and cosines typically converge faster. Piecewise functions, like rectangle pulses, reveal truncation artifacts more clearly.