AE2Anexampleofperiodicextensionforahalf-rangeseries.PDFVIP

AE2: An example of periodic extension for a half-range series Recall that for a function f (x) defined on [0 , L], which is extended as an even periodic function, has a Fourier series representation (bn = 0) ∞ L L f (x) = 1 a0 + an cos nπx ; an = 2 f (x) cos nπx dx ; a0 = 2 f (x) dx 2 L L 0 L L 0 n=1 whereas if it is extended to be an odd periodic function, its Fourier series representation is (an = 0) ∞ L f (x) = bn sin nπx ; bn = 2 f (x) sin nπx dx . L L 0 L n=1 We are given the “tent-function” x 0 ≤ x ≤ 1 π f (x) = 1 2 (1) π − x 2 π ≤ x ≤ π 1) Even extension of f (x) defined in (1) with L = π : x 1 1 −π − 2 π 2 π π Because this extension is even by construction, bn = 0. To calculate an : 1 πa0 = π f (x)dx = π/2 xdx + π (π − x)dx = 1 π2 a0 = 1 π (2) 2 4 2 0 0 π/2 1 πan = π f (x)dx = π/2 x cos nx dx + π (π − x) cos nx dx