MIT18_03S10_7ex

MIT18_03S10_7ex - 7 Fourier Series Based on exercises in...

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Unformatted text preview: 7. Fourier Series Based on exercises in Chap. 8, Edwards and Penney, Elementary Differential Equations 7A. Fourier Series 7A-1. Find the smallest period for each of the following periodic functions: a) sin t/ 3 b) | sin t | c) cos 2 3 t 7A-2. Find the Fourier series of the function f ( t ) of period 2 which is given over the interval − < t by , − < t 0; − t, − < t < 0; a) f ( t ) = b) f ( t ) = 1 , < t t, t , m = n ; 7A-3. Give another proof of the orthogonality relations cos mt cos nt dt = − , m = n . 1 by using the trigonometric identity: cos A cos B = 2 cos( A + B ) + cos( A − B ) . 7A-4. Suppose that f ( t ) has period P . Show that I f ( t ) dt has the same value over any interval I of length P , as follows: a + P a a) Show that for any a , we have f ( t ) dt = f ( t ) dt . (Make a change of variable.) P P a + P b) From part (a), deduce that f ( t ) dt = f ( t ) dt . a 7B. Even and Odd Series; Boundary-value Problems 7B-1. a) Find the Fourier cosine series of the function...
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MIT18_03S10_7ex - 7 Fourier Series Based on exercises in...

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