Mathematic Methods HW Solutions 35

Mathematic Methods HW Solutions 35 - Chapter 7 35 9.18...

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Unformatted text preview: Chapter 7 35 9.18 continued Function of period 3: √ 3 1 an = nπ sin 2nπ = 2nπ {1, −1, 0, and repeat}, a0 /2 = 1/3 3 1 3 bn = nπ (1 − cos 2nπ ) = 2nπ {1, 1, 0, and repeat} 3 √ 3 πx 1 + 2π (cos 2πx − 1 cos 4πx + 1 cos 8πx − 1 cos 103 3 3 2 3 4 3 5 1 πx + 23 (sin 2πx + 1 sin 4πx + 1 sin 8πx + 5 sin 103 · · · ) π 3 2 3 4 3 ···) fp (x) = ∞ 4 2 (−1)n cos 2nx 9.19 fc (x) = fp (x) = − π π1 4 n2 − 1 0, n even 2 For fs , bn = π 2 , n = 1 + 4k n+1 2 n−1 , n = 3 + 4k 1 1 2 fs (x) = π (sin x + sin 3x + 3 sin 5x + 3 sin 7x + 9.20 fc (x) = 2 fs (x) = π fp (x) = ∞ 1 4 +2 3π 8 (−1)n+1 sin nπx − 3 n π 1 1 1 + 3 π2 ∞ 1 1 cos 2nπx − n2 π 1 4 1 9.21 fc (x) = fp (x) = − 2 2π fs (x) = 8 π2 sin sin 9x + 1 5 sin 11x · · · ) (−1)n cos nπx n2 1 ∞ 1 5 ∞ 1 odd n ∞ 1 odd n ∞ 1 1 sin nπx n3 1 sin 2nπx n 1 cos nπx n2 πx 1 3πx 1 5πx 8 nπ − 2 sin + 2 sin · · · ; bn = 2 2 sin 2 3 2 5 2 nπ 2 20 9.22 Even function: an = − nπ sin nπ 2 πx πx 20 fc (x) = 15 − π (cos 20 − 1 cos 320 + 3 1 5 πx cos 520 · · · ) Odd function: 20 20 bn = nπ (cos nπ + 1 − 2 cos nπ ) = nπ {3, −2, 3, 0, and repeat} 2 πx πx πx fs (x) = 20 (3 sin πx − 2 sin 220 + 3 sin 320 + 3 sin 520 · · · ) π 20 2 3 5 Function of period 20: ∞ 20 1 nπx fp (x) = 15 − sin π1n 10 odd n 9.23 f (x, 0) = 8h π2 sin 1 3πx 1 5πx πx − 2 sin + 2 sin ··· l 3 l 5 l ∞ nπx λn 8h sin where 9.24 f (x, 0) = 2 π n2 l 1 √ √ √ λ1 = 2 − 1, λ2 = 2, λ3 = 2 + 1, λ4 = 0, λ5 = −( 2 + 1), √ λ6 = −2, λ7 = − 2 + 1, λ8 = 0, ..., λn = 2 sin nπ − sin nπ 4 2 9.26 f (x) = 1 48 − 2 π4 ∞ 1 odd n cos nπx n4 9.27 f (x) = 8π 4 − 48 15 ∞ 1 (−1)n cos nx n4 ...
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This note was uploaded on 02/29/2012 for the course MHF 2312 taught by Professor Dr.chet during the Fall '11 term at UNF.

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