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s6_95-100a

# s6_95-100a - (a(10 pts Z 1 ±1 e ax e x 1 dx = ² sin...

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ACM95a/100a November 11, 2009 Problem Set VI Please deposit your completed homework set to by the due-time in the Firestone 303 door-slot. Please write your Grading Section Number at the top of the ±rst page of your homework set. 1. (20 pts.) Use contour integration to evaluate the integrals (a) (10 pts.) Z 2 ± 0 2 + sin( ± ) ; (b) (10 pts.) Z ± ± ± cos( ) 1 ± 2 a cos( ± ) + a 2 ( j a j < 1) : For the second integral, assume n is a non-negative integer. 2. (30 pts.) Use contour integration to evaluate the integrals (a) (10 pts.) Z 1 ±1 dx 1 + x 4 , (b) (10 pts.) Z 1 ±1 x 2 dx (1 + x 2 ) 2 , (c) (10 pts.) Z 1 ±1 cos( x ) (1 + x 2 ) dx . 3. (20 pts.) For 0 < a < 1 show that
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Unformatted text preview: (a) (10 pts.) Z 1 ±1 e ax e x + 1 dx = ² sin( ²a ) , (b) (10 pts.) Z 1 ±1 cosh( ax ) cosh( x ) dx = ² sec ²a 2 . [Hint: For the ±rst integral integrate exp( az ) = (exp( z ) + 1) around a rectangle with vertices at ± R , R , R +2 ²i and ± R +2 ²i . The second integral can be obtained similarly, or deduced from the ±rst.] 4. (30 pts.) Obtain the formulae (a) (15 pts.) Z 1 ±1 1 ± cos( x ) x 2 dx = ² , (b) (15 pts.) Z 1 sin( ²x ) x (1 ± x 2 ) dx = ² . Due at 5pm on Wed. November 18....
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