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# Quiz5-2 - 2 3 2 2 lim lim lim 1 1 2 4 8 4 4 4 iwz iwz iwz...

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EGM 4313 Quiz 5 NAME:_____________________________ Spring 2010 Open book, closed notes, no calculators. Evaluate ( 29 ( 29 2 2 cos 1 wx dx x -∞ + . Consider ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 0 cos sin sin cos 2 2 2 2 2 2 2 2 2 2 0 0 cos sin 1 1 1 1 cos cos 1 1 1 1 i R R iwz iwRe i i C R R iwR i R R wR iwR i i i i R R wx wx e e dz dx i dx Rie d z x x R e wx wx e e dx Rie d dx Rie d x R e x R e θ π θ θ θ θ π π θ θ θ θ θ θ θ θ θ - - + - + - - = + + + + + + = + = + + + + + Ñ The second integral in the first line vanishes (odd function). The last integral will vanish as R →∞ if w is greater than or equal to zero. Since ( 29 ( 29 cos cos wx wx - = we can use the result for w> 0 and extend as an even function. ( 29 2 2 1 iwz e z + has a second order pole at z=i . Hence we find the residue by
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Unformatted text preview: 2 3 2 2 lim lim lim 1 1 2 4 8 4 4 4 iwz iwz iwz iwz z i z i z i w w w w w d e d e iwe e z i dz dz z i z i z i z w e iwe e we e i i i i → → →----- -= =- + + + + + =-= + =--Therefore ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 cos 1 1 2 4 2 1 1 w w iwz C wx w e w e e dz dx i i z x--∞-∞ + + = = = + + ∫ ∫ Ñ for w >0. To make this an even function ( 29 ( 29 ( 29 2 2 1 cos 2 1 w w e wx dx x-∞-∞ + = + ∫...
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