Fourier Coefficient Calculator
Calculate Fourier series coefficients and evaluate series approximations for periodic functions.
Input
Square wave: f(x) = sgn(sin(πx/L))
Period = 6.2832, L = 3.1416
Fourier Series at x = 1.5708
1.0630539691
Actual: 1.000000, Error: 6.3054e-2
Fourier Coefficients
| n | aₙ | bₙ | |cₙ| |
|---|---|---|---|
| 0 | 0.000000 | 0.000000 | 0.000000 |
| 1 | 0.000000 | 1.273240 | 1.273240 |
| 2 | 0.000000 | 0.000000 | 0.000000 |
| 3 | 0.000000 | 0.424413 | 0.424413 |
| 4 | 0.000000 | 0.000000 | 0.000000 |
| 5 | 0.000000 | 0.254648 | 0.254648 |
| 6 | 0.000000 | 0.000000 | 0.000000 |
| 7 | 0.000000 | 0.181891 | 0.181891 |
| 8 | 0.000000 | 0.000000 | 0.000000 |
| 9 | 0.000000 | 0.141471 | 0.141471 |
| 10 | 0.000000 | 0.000000 | 0.000000 |
Convergence
S_1=1.273S_2=1.273S_3=0.849S_4=0.849S_5=1.103S_6=1.103S_7=0.922S_8=0.922S_9=1.063S_10=1.063
Parseval's Theorem
Σ(aₙ² + bₙ²) ≈ 1.919210
Fourier Series
Formula
f(x) = a₀/2 + Σ(aₙcos(nπx/L) + bₙsin(nπx/L))
Applications
- Signal processing
- Heat equation solutions
- Vibration analysis