Even & Odd Functions, Half-Range Sine/Cosine Series & Parseval's Identity

Explanation

Half-range expansions solve boundary value problems in heat conduction and wave vibration over finite rods [0, L] with specified Dirichlet (temperature zero at ends => Sine series) or Neumann (insulated ends => Cosine series) boundary conditions. Parseval's identity is the trigonometric generalization of the Pythagorean theorem in function space, stating that total signal energy equals the sum of energy across all spectral harmonics.

Key points

• Half-Range Expansions: Given f(x) on [0, L], we can extend it as an even function to get a Half-Range Cosine Series, or as an odd function to get a Half-Range Sine Series.
• Half-Range Cosine Series: f(x) = a₀/2 + ∑_{n=1}^∞ a_n cos(nπx/L) where a₀ = (2/L) ∫_0^L f(x) dx, a_n = (2/L) ∫_0^L f(x) cos(nπx/L) dx.
• Half-Range Sine Series: f(x) = ∑_{n=1}^∞ b_n sin(nπx/L) where b_n = (2/L) ∫_0^L f(x) sin(nπx/L) dx.
• Parseval's Identity (Root Mean Square Energy): (1/L) ∫{-L}^L [f(x)]² dx = a₀²/2 + ∑{n=1}^∞ (a_n² + b_n²).
• For Half-Range Cosine: (2/L) ∫0^L [f(x)]² dx = a₀²/2 + ∑{n=1}^∞ a_n².
• For Half-Range Sine: (2/L) ∫0^L [f(x)]² dx = ∑{n=1}^∞ b_n².

Example

Code examples

// Application of Parseval's Identity to Sum 1/n^4:
// Consider Half-Range Cosine Series of f(x) = x on [0, π]:
// f(x) = π/2 - (4/π) ∑_{n=1,3,5...} [cos(nx) / n^2]
// Here: a0/2 = π/2 => a0 = π
// For odd n: an = -4 / (π n^2); For even n: an = 0.
//
// By Parseval's Identity for Half-Range Cosine Series:
//   (2/π) ∫_0^π [f(x)]^2 dx = a0^2 / 2 + ∑_{n=1}^∞ an^2
//
// LHS = (2/π) ∫_0^π x^2 dx = (2/π) [ x^3 / 3 ]_0^π = (2/π) (π^3 / 3) = 2π^2 / 3
// RHS = (π^2 / 2) + ∑_{odd n} [ -4 / (π n^2) ]^2
//     = (π^2 / 2) + (16 / π^2) ∑_{odd n} (1 / n^4)
//
// Equating LHS and RHS:
//   2π^2 / 3 - π^2 / 2 = (16 / π^2) ∑_{odd n} (1 / n^4)
//   π^2 / 6 = (16 / π^2) ∑_{odd n} (1 / n^4)
//   ∑_{n=1,3,5...}^∞ (1 / n^4) = π^4 / 96 (Q.E.D.!)

Exam questions

• [8 Marks] Find the half-range sine series of f(x) = x (π - x) in 0 < x < π. Hence deduce that 1/1³ - 1/3³ + 1/5³ - ... = π³ / 32.
• [8 Marks] Using Parseval's identity for f(x) = x² on (-π, π), prove that ∑_{n=1}^∞ (1 / n⁴) = π⁴ / 90.
• [7 Marks] Expand f(x) = l - x in a half-range cosine series in the interval 0 < x < l.