Fourier Series
Learn the Fourier series formula for decomposing periodic functions into sine and cosine components, with worked examples.
The Formula
The Fourier series is a mathematical technique that expresses any periodic function as a sum of simple sine and cosine waves. Developed by the French mathematician Joseph Fourier in 1807 while studying heat conduction, this formula became one of the most influential discoveries in all of mathematics and engineering.
The idea is beautifully simple. Any repeating pattern, no matter how complex or jagged, can be broken down into a combination of smooth, regular waves. Each wave has a specific frequency (how fast it oscillates) and amplitude (how tall it is). By adding enough of these waves together, you can reconstruct the original function with perfect accuracy.
The coefficients a₀, aₙ, and bₙ determine how much of each wave component is present in the original function. These are calculated using integrals over one full period of the function. The coefficient a₀ represents the average value of the function. The cosine coefficients aₙ capture the symmetric parts, while the sine coefficients bₙ capture the antisymmetric parts.
Fourier series are fundamental to modern technology. Digital music, image compression (JPEG), signal processing, telecommunications, and medical imaging (MRI) all rely on Fourier analysis. When you stream music or make a phone call, Fourier mathematics is working behind the scenes to encode and decode the signals. Engineers use Fourier series to analyze vibrations in structures, electrical engineers use them to study alternating current circuits, and physicists use them to solve partial differential equations like the heat equation and wave equation.
Variables
| Symbol | Meaning |
|---|---|
| f(x) | The periodic function being decomposed |
| L | Half the period of the function (period = 2L) |
| a₀ | The constant term (twice the average value of f(x) over one period) |
| aₙ | Cosine coefficients: (1/L) ∫ f(x) cos(nπx/L) dx over one period |
| bₙ | Sine coefficients: (1/L) ∫ f(x) sin(nπx/L) dx over one period |
| n | The harmonic number (1, 2, 3, ...) indicating each wave's frequency |
Example 1: Square Wave
Problem: Find the Fourier series for a square wave that equals +1 for 0 < x < π and −1 for −π < x < 0.
The function is odd, so all cosine coefficients aₙ = 0 and a₀ = 0.
bₙ = (1/π) ∫ from −π to π of f(x) sin(nx) dx = 2/(nπ) for odd n, and 0 for even n.
f(x) = (4/π)[sin(x) + sin(3x)/3 + sin(5x)/5 + sin(7x)/7 + ...]
The square wave is built from odd harmonics with amplitudes that decrease as 1/n.
Example 2: Sawtooth Wave
Problem: Find the Fourier series for f(x) = x on the interval (−π, π).
The function is odd, so a₀ = 0 and all aₙ = 0.
bₙ = (1/π) ∫ from −π to π of x sin(nx) dx = 2(−1)ⁿ⁺¹ / n
f(x) = 2[sin(x) − sin(2x)/2 + sin(3x)/3 − sin(4x)/4 + ...]
The sawtooth wave uses all harmonics with alternating signs and amplitudes of 2/n.
When to Use It
The Fourier series is essential whenever you work with periodic signals or need to analyze frequency content.
- Signal processing and audio engineering — decomposing sound into frequency components
- Electrical engineering — analyzing AC circuits and waveforms
- Image compression — JPEG and other formats use related Fourier techniques
- Solving partial differential equations — heat equation, wave equation, vibration problems
- Vibration analysis — identifying resonant frequencies in mechanical structures