← Back 3 Fourier DecompositionFourier's radical claim, initially met with skepticism from Lagrange and Laplace, was that any periodic function can be decomposed into a sum of sines and cosines.
Let f(x) be periodic with period 2pi. Then:
f ( x ) = a 0 2 + ∑ n = 1 ∞ ( a n cos n x + b n sin n x ) f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\bigl(a_n \cos nx + b_n \sin nx\bigr) f ( x ) = 2 a 0 + n = 1 ∑ ∞ ( a n cos n x + b n sin n x ) The coefficients are determined by orthogonality:
a n = 1 π ∫ − π π f ( x ) cos ( n x ) d x , b n = 1 π ∫ − π π f ( x ) sin ( n x ) d x a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos(nx)\,dx, \quad b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin(nx)\,dx a n = π 1 ∫ − π π f ( x ) cos ( n x ) d x , b n = π 1 ∫ − π π f ( x ) sin ( n x ) d x Example: the square wave f(x) = sgn(sin x) decomposes as:
f ( x ) = 4 π ( sin x + sin 3 x 3 + sin 5 x 5 + sin 7 x 7 + ⋯ ) f(x) = \frac{4}{\pi}\left(\sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \frac{\sin 7x}{7} + \cdots\right) f ( x ) = π 4 ( sin x + 3 sin 3 x + 5 sin 5 x + 7 sin 7 x + ⋯ ) Only odd harmonics appear; each successive term sharpens the approximation.
f ( x ) = 4 π ∑ k = 0 ∞ sin ( ( 2 k + 1 ) x ) 2 k + 1 \boxed{\;f(x) = \frac{4}{\pi}\sum_{k=0}^{\infty} \frac{\sin\bigl((2k+1)x\bigr)}{2k+1}\;} f ( x ) = π 4 k = 0 ∑ ∞ 2 k + 1 sin ( ( 2 k + 1 ) x ) Fourier analysis underpins signal processing, quantum mechanics, and the solution of partial differential equations.