Half range cosine series: x

The half range cosine series on (0, L) is f(x) = a₀/2 + Σ aₙ cos(nπx/L), where a₀ = (2/L)∫₀ᴸ f dx and aₙ = (2/L)∫₀ᴸ f cos(nπx/L) dx. Here L = 2 (the function is extended as an even function).

Coefficients for f(x) = x:

a₀ = (2/2)∫₀² x dx = [x²/2]₀² = 2

aₙ = ∫₀² x cos(nπx/2) dx
   = [ x·(2/nπ) sin(nπx/2) ]₀² − (2/nπ)∫₀² sin(nπx/2) dx
   = 0 + (2/nπ)(2/nπ)[cos(nπx/2)]₀²
   = (4/n²π²)(cos nπ − 1)
   = 4((−1)ⁿ − 1)/(n²π²)

• n even: aₙ = 0
• n odd: aₙ = −8/(n²π²)

Series

x = 1 − (8/π²) Σ (n odd) cos(nπx/2) / n², 0 < x < 2

= 1 − (8/π²)[ cos(πx/2) + cos(3πx/2)/3² + cos(5πx/2)/5² + … ]

Check: at x = 0 the series gives 1 − (8/π²)(1 + 1/9 + 1/25 + …) = 1 − (8/π²)(π²/8) = 0.