Full Fourier Series:
f(x) = a₀ + Σ [aₙcos(nπx/L) + bₙsin(nπx/L)]
Coefficients:
a₀ = (1/2L) ∫₋ₗᴸ f(x) dx
aₙ = (1/L) ∫₋ₗᴸ f(x) cos(nπx/L) dx
bₙ = (1/L) ∫₋ₗᴸ f(x) sin(nπx/L) dx
Full Series
Contains both sine and cosine terms
Sine Series (odd)
Only sine terms, bₙ coefficients
Cosine Series (even)
Only cosine terms, a₀ and aₙ coefficients
Supported operations:
- +, -, *, /, ^ (power)
- sin(x), cos(x), tan(x)
- exp(x), ln(x), sqrt(x)
- abs(x), pi, e
Examples:
- x (sawtooth wave)
- x^2 (parabola)
- abs(x) (triangle wave)
- sin(x) + cos(2*x)
A Fourier series is a mathematical tool that represents a periodic function as an infinite sum of sine and cosine functions. Named after French mathematician Joseph Fourier, this powerful concept allows us to decompose complex periodic signals into simpler harmonic components. The Fourier series is fundamental in signal processing, physics, engineering, and many other fields where periodic phenomena need to be analyzed or synthesized.
The key insight is that any reasonably well-behaved periodic function can be expressed as a weighted sum of sinusoids at different frequencies. The weights (Fourier coefficients) determine how much each harmonic contributes to the overall shape of the function.
Fourier series have wide-ranging applications across science and engineering. In signal processing, they are used to analyze audio signals, filter noise, and compress data. In electrical engineering, they help analyze AC circuits and design filters. In physics, Fourier analysis is essential for studying heat conduction, wave propagation, and quantum mechanics.
Signal Processing
Audio analysis, image compression, filtering
Communications
Modulation, bandwidth analysis, spectral analysis
Physics & Engineering
Heat transfer, vibration analysis, quantum mechanics
Fourier series calculations follow standard mathematical formulas. Results depend on correct function input, period, and number of terms. This calculator uses numerical integration (Simpson's rule) which provides approximations. For exact analytical solutions, manual calculation or symbolic computation software is recommended. Always verify critical calculations with additional methods.