# 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 + ··· . Watson’s 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 ε ∼ x , global where x = O ( 1 ) and local when ε x = O ( 1 ) . Divide the range at δ and M where ε δ 1 and 1 M ε- 1 . Then I L 1 = Z δ / ε d u ( 1 + u )( 1 + ε 2 u ) = Z δ / ε d u 1 + u [ 1- ε 2 u + O ( ε 4 u 2 )] = [ log ( 1 + u )] δ / ε- ε 2 [ u- log ( 1 + u )] δ / ε + O ( ε 2 δ 2 ) = log δ ε + ε δ- εδ + ε 2 log δ ε + O ε 2 δ 2 , ε 3 δ , ε 2 δ 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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