M312 #002 Sp17
Partial Differential Equations – Homework #6
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Homework #6 – Fourier Series
1. Consider the function 𝑓(𝑥) = 𝑥, 0 < 𝑥 < 1.
(a) Graph the odd periodic extension of 𝑓(𝑥) on the interval −4 < 𝑥 < 4 and
compute its Fourier Sine series. Determine the period of this function.
(b) Graph the even periodic extension of 𝑓(𝑥) on the interval −4 < 𝑥 < 4 and
compute its Fourier Cosine series. Determine the period of this function.
(c) Assuming 𝑓(𝑥) is 1-periodic, graph the function on the interval −2 < 𝑥 < 2 and
compute its Full Fourier series.
2. Consider the function 𝑓(𝑥) = cos 𝑥, 0 < 𝑥 < 𝜋.
(a) Graph the odd periodic extension of 𝑓(𝑥) on the interval −2𝜋 < 𝑥 < 2𝜋 and
compute its Fourier Sine series. Determine the period of this function.
(b) Graph the even periodic extension of 𝑓(𝑥) on the interval −2𝜋 < 𝑥 < 2𝜋 and
compute its Fourier Cosine series. Determine the period of this function.
(c) Assuming 𝑓(𝑥) is 𝜋-periodic, graph the function on the interval −2𝜋 < 𝑥 < 2𝜋
and compute its Full Fourier series.
3. Compute the Fourier Sine series of 𝑓(𝑥) = 𝑥 2 − 𝐿𝑥, 0 < 𝑥 < 𝐿.
4. Compute the Full Fourier series of 𝑓(𝑥) = cos 3 𝑥, −𝜋 < 𝑥 < 𝜋.
5. Compute the Full Fourier series of the 2𝜋-periodic function 𝑓(𝑥) = sin3 (2𝑥) cos 2 (3𝑥).
6. Compute the Full Fourier series of the 2𝜋-periodic function 𝑓(𝑥) =
sin(𝑥) sin(2𝑥) sin(3𝑥).
7. Compute the Fourier Sine series of 𝑓(𝑥) = {
𝑥,
0<𝑥<
𝜋
𝜋 − 𝑥,
8. Compute the Fourier Cosine series of 𝑓(𝑥) = {
1,
0,
2
2
2
<𝑥<𝜋
0<𝑥<
𝜋
𝜋
for 0 < 𝑥 < 𝜋.
𝜋
2
<𝑥<𝜋
for 0 < 𝑥 < 𝜋.
−1, −1 < 𝑥 < 0
9. Consider the function 𝑓(𝑥) = {
for −1 < 𝑥 < 1.
1,
0<𝑥<1
(a) Assuming 𝑓(𝑥) is 2-periodic, compute its Full Fourier series.
1
1
1
(b) Use the result in part (a) to compute the sum: 1 − 3 + 5 − 7 + ⋯.
10. Consider the function 𝑓(𝑥) = |𝑥|, −1 < 𝑥 < 1.
(a) Assuming 𝑓(𝑥) is 2-periodic, compute its Full Fourier series.
1
1
1
(b) Use the result in part (a) to compute the sum: 1 + 9 + 25 + 49 + ⋯.