s09_mthsc208_hw21

s09_mthsc208_hw21 - MTHSC 208 (Differential Equations) Dr....

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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.

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