Math Help

Example: Find the Fourier Series representation of f(t) where

f(t) = and for help with this problem.

t² 0 ≤ t ≤ π

The Fourier series expansion of f(t) is:

∞

f(t) = a_o /2 + ∑ a_n cos nt + b_n sin nt

n = 1

Now calculate the Fourier coefficients a_o, a_n, b_n

Note: The period of f(t) is 2π and f(t) is zero from ˗ π ≤ t ≤ 0 . So one needs to

only evaluate the integrals from 0 ≤ t ≤ π.

π π π

a_o = (1/ π) ∫ f(t) dt = (1/ π) ∫ t² dt = (1/ π) t³/3 | = π²/3

0 0 0

π π

a_n = (1/ π) ∫ f(t) cos nt dt = (1/ π) ∫ t² cos nt dt now integrate by parts:

0 0

u = (1/ π) t² dv = cos nt dt

du = (2t/π) dt v = (1/n) sin nt

π π π

a_n = (1/ π) [ ( t²/n ) sin nt ] - (2/ nπ) ) ∫ t sin nt dt = (-2/ nπ) ) ∫ t sin nt dt

0 0 0

u = 2t/π dv = -sin nt dt

du = 2dt/ π v = (1/n) cos nt

π π π

a_n = [ 2t/n² cos nt ] - (2/ n²π) ∫ cos nt dt = [ (2t/n²π) cos nt ]

0 0 0

a_n = 2 cos nπ / n²

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