Exercise 7
1. Find the Fourier series for each of the following 2π-periodic functions.
(a) f(x) =
(x |x|< π/2
0 |x|> π/2, p= 2π (b) f(x) =
(0 −π < x <0
1 06t < π , p= 2π (c) f(x) =
(0 −π < x <0
x 0< x < π , p= 2π (d) f(x) =x2 for |x|< π, p= 2π
2. Find the Fourier series for each of the following 2L-periodic functions.
(a) f(x) =
(−x −< x <0 x 0< x <1 . (b) f(x) =
(−4−x −4< x < 0 4−x 0< x < 4
(c) f(x) = cos(πx), (−12 < x < 12), p= 1
3. In each of the following periodic functions, use the convergence theorem to determine the sum of the Fourier series of the function.
(a) f(x) =
2x −3< x <−2 0 −2< x <1 x2 1< x <3
, p= 6
(b) f(x) =
(2x−2 −π < x <1
3 1< x < π , p= 2π (c) f(x) =
(x2 −π < x <0
2 0< x < π , p= 2π
4. Use the Fourier series expansion of f(x) = x2, (−1< x <1), p= 2, to show that 1 + 1
4+ 1 16+ 1
25 +· · ·= π2 6
1
