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Unformatted text preview: MTHSC 208 (Diﬀerential Equations)
Dr. Matthew Macauley
HW 21
Due Friday April 10th, 2009
(1) Compute the complex Fourier series for the function deﬁned on the interval [−π, π ]:
f (x) = −1,
4, −π ≤ x < 0,
0 ≤ x ≤ π. Use the cn ’s to ﬁnd the coeﬃcients of the real Fourier series (the an ’s and bn ’s).
(2) Find the real and complex Fourier series for the function deﬁned on the interval [−π, π ]:
f (x) = 0,
1, −π ≤ x < 0,
0 ≤ x ≤ π. Only compute one of these directly (your choice), and then use the formulas relating the
real and complex coeﬃcients to compute the other.
(3) Compute the complex Fourier series for the function f (x) = π − x deﬁned on the interval
[−π, π ]. Use the cn ’s to to ﬁnd the coeﬃcients of the real version of the Fourier series.
(4) Prove Parseval’s identity:
1
π ∞ π (f (x))2 dx =
−π 12
(a2 + b2 ) .
a+
n
2 0 n=1 n (5) Use Parseval’s identity, and the Fourier series of the function f (x) = x2 on [−π, π ], to
∞
1
compute
.
n4
n=1
∞ (6) Compute
at f (π ). 1
. Hint : Compute the Fourier series for f (x) = x, and then look
(2n + 1)2
n=1 ...
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This note was uploaded on 03/12/2012 for the course MTHSC 208 taught by Professor Staufeneger during the Spring '09 term at Clemson.
 Spring '09
 Staufeneger
 Differential Equations, Equations, Fourier Series

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