MA212: Assignment #9
Required Reading:
• Sections 12.2-12.3
To be turned in May 17th at the start of class.
1. Textbook §12.2 #s 5, 9, 12, 17
2. Textbook §12.3 #s 11, 12, 18, 20, 28, 39, 40
3. Let f (x) = x on [−π, 0]
(a) Sketch the even periodic extension of f (x) to the interval [−2π, 2π]. Call this fe (x).
(b) Find a Fourier series expansion for fe (x) on [−π, π].
(c) Sketch the odd periodic extension of f (x) to the interval [−2π, 2π]. Call this fo (x).
(d) Find a Fourier series expansion for fo (x) on [−π, π].
4. On the interval [−1, 1], define
f (x) = sin(πx) + a sin(ωx)
If ω π and a 1, then we can think of a sin(ωx) as high frequency low amplitude noise added to our signal.
(a) Prove that the Fourier Series for f (x) is
∞
X
2nπa(−1)n sin ω
2πa sin ω
sin(πx)
+
sin(nπx)
1− 2
2
ω −π
ω 2 − (nπ)2
n=2
(b) Set a = 0.1 and ω = 62. Plot the absolute value of the Fourier coefficients for n = 1 to n = 40. Specifically,
use Maple’s pointplot to plot n on the horizontal axis and |bn | on the vertical axis. Which Fourier coefficient
seems to correspond most strongly to the noise? Which corresponds to the signal?
(c) Plot the first 100 terms in the Fourier series. Plot f (x). Compare the two plots.
(d) In part b), you found a value for n for which bn seemed to correspond to the noise. Set this coefficient to 0.
Plot the first 100 terms in the Fourier series (remember, you’ve zeroed one of the terms). Plot the original
function. Compare the two plots. Contrast with part c).
5. If n and m are integers, prove that
Z
p
einπx/p e−imπx/p dx =
−p
2p
0
if n = m
if n =
6 m
Use this result to compute the coefficients in the following series:
x=
∞
X
cn einπx/p
n=−∞
Euler’s formula is useful for simplifying your answers (eiθ = cos θ + i sin θ). For example, eπi = −1 and e2πi = 1.
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