sol6 - MAE294B/SIO203B: Methods in Applied Mechanics Winter...

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Unformatted text preview: MAE294B/SIO203B: Methods in Applied Mechanics Winter Quarter 2010 http://maecourses.ucsd.edu/mae294b Solution VI 1 Integrate by parts: E 1 ( x ) =- e- t t x- Z x e- t t 2 d t = e- x x-- e- t t 2 x + Z x 2 e- t t 2 d t = e- x 1 x- 1 x 2 +- 2 e- t t 2 x + Z x 6 e- t t 2 d t = e- x 0! x- 1! x 2 + 2! x 3 + . Watsons lemma: make the change of variable t = x ( 1 + u ) and use ( 1 + u )- 1 = n = 1 (- u ) n . Then E 1 ( x ) = Z e- x ( 1 + u ) 1 + u d u e- x n = (- 1 ) n n ! x n + 1 . 2 There are three regions: local when x , global where x = O ( 1 ) and local when x = O ( 1 ) . Divide the range at and M where 1 and 1 M - 1 . Then I L 1 = Z / d u ( 1 + u )( 1 + 2 u ) = Z / d u 1 + u [ 1- 2 u + O ( 4 u 2 )] = [ log ( 1 + u )] / - 2 [ u- log ( 1 + u )] / + O ( 2 2 ) = log + - + 2 log + O 2 2 , 3 , 2 2 is the first local contribution. The global contribution isis the first local contribution....
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sol6 - MAE294B/SIO203B: Methods in Applied Mechanics Winter...

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