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Mathematic Methods HW Solutions 35

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

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Chapter 7 35 9.18 continued Function of period 3: a n = 1 sin 2 3 = 3 2 { 1 , - 1 , 0 , and repeat } , a 0 / 2 = 1 / 3 b n = 1 (1 - cos 2 3 ) = 3 2 { 1 , 1 , 0 , and repeat } f p ( x ) = 1 3 + 3 2 π (cos 2 πx 3 - 1 2 cos 4 πx 3 + 1 4 cos 8 πx 3 - 1 5 cos 10 πx 3 · · · ) + 3 2 π (sin 2 πx 3 + 1 2 sin 4 πx 3 + 1 4 sin 8 πx 3 + 1 5 sin 10 πx 3 · · · ) 9.19 f c ( x ) = f p ( x ) = 2 π - 4 π summationdisplay 1 ( - 1) n cos 2 nx 4 n 2 - 1 For f s , b n = 2 π 0 , n even 2 n +1 , n = 1 + 4 k 2 n - 1 , n = 3 + 4 k f s ( x ) = 2 π (sin x + sin 3 x + 1 3 sin 5 x + 1 3 sin 7 x + 1 5 sin 9 x + 1 5 sin 11 x · · · ) 9.20 f c ( x ) = 1 3 + 4 π 2 summationdisplay 1 ( - 1) n n 2 cos nπx f s ( x ) = 2 π summationdisplay 1 ( - 1) n +1 n sin nπx - 8 π 3 summationdisplay 1 odd n 1 n 3 sin nπx f p ( x ) = 1 3 + 1 π 2 summationdisplay 1 1 n 2 cos 2 nπx - 1 π summationdisplay 1 1 n sin 2 nπx 9.21 f c ( x ) = f p ( x ) = 1 2 - 4 π 2 summationdisplay 1 odd n 1 n 2 cos nπx f s ( x ) = 8 π 2 parenleftbigg sin πx 2 - 1 3 2 sin 3 πx 2 + 1 5 2 sin 5 πx 2 · · · parenrightbigg ; b n = 8 n 2 π 2 sin 2 9.22 Even function: a n = - 20 sin 2 f c ( x ) = 15 - 20 π (cos πx 20 - 1 3 cos 3 πx 20 + 1 5 cos 5 πx 20 · · · ) Odd function: b n = 20 (cos 2 + 1 - 2 cos ) = 20 { 3 , - 2 , 3 , 0 , and repeat } f s ( x ) = 20 π (3 sin πx 20 - 2 2 sin 2 πx 20 + 3
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