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Mathematic Methods HW Solutions 32

# Mathematic Methods HW Solutions 32 - Chapter 7 8.3 32 1 an...

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Chapter 7 32 8.3 a n = - 1 sin 2 , a 0 / 2 = c 0 = 1 4 b n = 1 ( cos 2 - cos ) = 1 { 1 , - 2 , 1 , 0 , and repeat } c n = i 2 ( e - inπ - e - inπ/ 2 ) = 1 2 {- (1 + i ) , 2 i, 1 - i, 0, and repeat } , n> 0; c - n = c n f ( x ) = 1 4 + 1 2 π bracketleftBig - (1 + i ) e iπx/l - (1 - i ) e - iπx/l + 2 i 2 ( e 2 iπx/l - e - 2 iπx/l ) + 1 - i 3 e 3 iπx/l + 1+ i 3 e - 3 iπx/l - 1+ i 5 e 5 iπx/l - 1 - i 5 e - 5 iπx/l · · · bracketrightBig = 1 4 - 1 π ( cos πx l - 1 3 cos 3 πx l + 1 5 cos 5 πx l · · · ) + 1 π ( sin πx l - 2 2 sin 2 πx l + 1 3 sin 3 πx l + 1 5 sin 5 πx l - 2 6 sin 6 πx l · · · ) 8.4 c 0 = a 0 / 2 = - 1 / 2; for n negationslash = 0, coefficients are 2 times the coefficients in Problem 8.3. f ( x ) = - 1 2 - 1 π bracketleftBig (1 + i ) e iπx/l + (1 - i ) e - iπx/l - 2 i 2 ( e 2 iπx/l - e - 2 iπx/l ) - 1 - i 3 e 3 iπx/l - 1+ i 3 e - 3 iπx/l + 1+ i 5 e 5 iπx/l + 1 - i 5 e - 5 iπx/l · · · bracketrightBig = - 1 2 - 2 π ( cos πx l - 1 3 cos 3 πx l + 1 5 cos 5 πx l · · · ) + 2 π ( sin πx l - 2 2 sin 2 πx l + 1 3 sin 3 πx l + 1 5 sin 5 πx l - 2 6 sin 6 πx l · · · ) 8.5 a n = - 2 sin 2 , a 0 = 0, b n = 1 (2 cos 2 - 1 - cos ) = - 4 { 0 , 1 , 0 , 0, and repeat } c n = 1 2 inπ
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