Problem Set #9
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1. Consider the 2π-periodic function f (x) = x2 on [0, 2π). Find the Fourier series of f (x).
x
.
2. Consider the function f (x) = sin
2
(a) Compute the Fourier coefficient a0 for this function.
(b) Use a right-hand Riemann sum with 4 rectangles to approximate the Fourier coefficient
a2 for this function.
(c) Explain why bk must be zero for all k.
∞
4
2 X
cos(kx). Use this to find the sum of
(d) The Fourier series for f (x) is +
π k=1 π(1 − 4k 2 )
∞
X
1
.
1 − 4k 2
k=1

−π

0 −π ≤ x < 2
3. Consider the 2π-periodic function f (x) such that f (x) = x −π
≤ x < π2
2


0 π2 ≤ x < π
(a) Graph, F (x), the Fourier series of f (x) for −3π ≤ x ≤ 3π.
∞
∞
X
X
2(−1)k+1
(−1)k+1
(b) f (x) has Fourier series F (x) =
sin
(2k
−
1)x
+
sin(2kx).
2
π(2k
−
1)
2k
k=1
k=1
∞
X
π
1
Plug x = into the Fourier series to find the sum of
.
2
2
(2k
−
1)
k=1
4. Suppose f (x) is a 2π-periodic such that:
•
lim f (x) = 1, f (−π) = 1,
x→−π −
• lim− f (x) = 2, f (0) = 2,
x→0
•
lim
x→(−π/2)−
∞
X
k=1
(a) f (20π)
(b) f (21π)
(c) F (20π)
lim f (x) = 8
x→0+
f (x) = 4, f (−π/2) = 4,
Let F (x) = 3 +
following:
lim f (x) = 1
x→−π +
ak cos(kx) +
∞
X
k=1
lim
x→(−π/2)+
f (x) = 4
bk sin(kx) be the Fourier series for f (x). Evaluate the
(d) F (21π)
∞
X
(e)
ak
k=1
(f)
∞
X
(−1)k ak
k=1
5. Let f (x) be a 2π-periodic function with Fourier series
∞
∞
X
X
2k π
(−1)k
cos((2k − 1)x) +
sin((2k − 1)x)
F (x) = π +
(2k
+
1)!
k!
k=1
k=1
Evaluate the following:
(a) F (0)
(b) F (π)
(c) F π2
Z π
(d)
f (x) 2 + cos(3x) − sin(5x) dx
−π