Fourier series of full-wave rectified sine

x(t) = |sin t| (amplitude 1), a periodic wave with period T₀ = π, so ω₀ = 2π/T₀ = 2 rad/s.

On 0 < t < π, x(t) = sin t. The wave is even, so only cosine terms exist (bₖ = 0).

Coefficients

a₀ = (1/π) ∫₀^π sin t dt = 2/π

aₙ = (2/π) ∫₀^π sin t cos(2nt) dt = (1/π) ∫₀^π [sin((1+2n)t) + sin((1-2n)t)] dt = (1/π) [ 2/(1+2n) + 2/(1-2n) ] = (1/π) · 4/(1 - 4n²)

aₙ = -4 / (π(4n² - 1))

Result

x(t) = 2/π - (4/π) Σ_{n=1}^{∞} cos(2nt) / (4n² - 1)

= 2/π - (4/3π) cos 2t - (4/15π) cos 4t - (4/35π) cos 6t - ...

The mean value is 2/π = 0.637.

Exponential form

cₙ = 2 / (π(1 - 4n²)): c₀ = 0.637, c₁ = -0.212, c₂ = -0.0424, c₃ = -0.0182 (verified numerically). Only even harmonics of the original sine (2, 4, 6, ...) are present.