pp2a_7 - 7 Define f(x on the interval by f(x = Find the...

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7. Define f ( x ) on the interval ( - π, π ) by: f ( x ) = ( x 2 if 0 x < π 0 if - π < x 0 Find the Fourier series of f ( x ) . Solution: a 0 = 1 π Z π - π f ( x ) dx = 1 π Z π 0 x 2 dx = 1 π 1 3 π 3 = π 2 3 For n N . a n = 1 π Z π - π f ( x ) cos( nx ) dx = 1 π Z π 0 x 2 cos( nx ) dx = 1 π x 2 sin( nx ) n + 2 x cos( nx ) n 2 - 2 sin( nx ) n 3 x = π x =0 = 2 π cos( ) n 2 = 2 π ( - 1) n n 2 1
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b n = 1 π Z π - π f ( x ) sin( nx ) dx = 1 π Z π 0 x 2 sin( nx ) dx = 1 π - x 2 cos( nx ) n + 2 x sin( nx ) n 2 + 2 cos( nx ) n 3 x = π x =0 = 1 π - π 2 cos( ) n + 2 π sin( ) n 2 + 2 cos( ) n 3 - 1 π - 0 2 cos( n 0) n + 2 · 0 sin( n 0) n 2 + 2 cos( n 0) n 3 = 1 π - π 2 cos( ) n + 2 cos( ) n 3 - 1 π 2 n 3 = - π cos( ) n + 2 cos( ) πn 3 - 2 πn
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