Fourier series: eˣ

f(x) = eˣ on (−π, π), period 2π. Use ∫ eˣ cos nx dx = eˣ(cos nx + n sin nx)/(1 + n²) and ∫ eˣ sin nx dx = eˣ(sin nx − n cos nx)/(1 + n²).

Coefficients

a₀ = (1/π) ∫_{−π}^{π} eˣ dx = (e^π − e^(−π))/π = 2 sinh π / π

aₙ = (1/π) [ eˣ (cos nx + n sin nx)/(1 + n²) ]_{−π}^{π}
   = (−1)ⁿ (e^π − e^(−π)) / (π(1 + n²)) = 2 (−1)ⁿ sinh π / (π (1 + n²))

bₙ = (1/π) [ eˣ (sin nx − n cos nx)/(1 + n²) ]_{−π}^{π}
   = −n (−1)ⁿ (e^π − e^(−π)) / (π(1 + n²)) = −2n (−1)ⁿ sinh π / (π (1 + n²))

(since cos nπ = (−1)ⁿ and sin nπ = 0.)

Series

eˣ = (sinh π / π) { 1 + 2 Σ_{n=1}^∞ [ (−1)ⁿ/(1 + n²) ] (cos nx − n sin nx) }

Expanded:

eˣ = (sinh π/π) [ 1 − cos x + sin x + (2/5)(cos 2x − 2 sin 2x) − (2/10)(cos 3x − 3 sin 3x) + ... ]

(sinh π/π ≈ 3.676.) At x = ±π the series gives the mean (e^π + e^(−π))/2 = cosh π.