s6_95-100a

s6_95-100a - (a) (10 pts.) Z 1 1 e ax e x + 1 dx = sin( a )...

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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 = sin( a ) , (b) (10 pts.) Z 1 1 cosh( ax ) cosh( x ) dx = sec a 2 . [Hint: For the rst integral integrate exp( az ) = (exp( z ) + 1) around a rectangle with vertices at R , R , R +2 i and R +2 i . 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 = , (b) (15 pts.) Z 1 sin( x ) x (1 x 2 ) dx = . Due at 5pm on Wed. November 18....
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This note was uploaded on 01/14/2010 for the course ACM 1 taught by Professor Prof during the Fall '09 term at Caltech.

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