7hw7 - Physics 315: Oscillations and Waves Homework 7: Due...

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Unformatted text preview: Physics 315: Oscillations and Waves Homework 7: Due in class on Wednesday, Nov. 4th 1. Suppose that F ( x ) = exp parenleftbigg- x 2 2 2 x parenrightbigg . Demonstrate that F ( k ) 1 2 integraldisplay F ( x ) e i k x dx = 1 radicalbig 2 2 k exp parenleftbigg- k 2 2 2 k parenrightbigg , where i is the square-root of minus one, and k = 1 / x . [Hint: You will need to complete the square of the exponent of e, transform the variable of integration, and then make use of the standard result that integraltext e y 2 dy = .] Hence, show from de Moives theorem, exp( i ) cos + i sin , that C ( k ) 1 2 integraldisplay F ( x ) cos( k x ) dx = 1 radicalbig 2 2 k exp parenleftbigg- k 2 2 2 k parenrightbigg , S ( k ) 1 2 integraldisplay F ( x ) sin( k x ) dx = 0 . 2. Demonstrate that integraldisplay 1 radicalbig 2 2 k exp parenleftbigg- k 2 2 2 k parenrightbigg dk...
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This note was uploaded on 11/02/2010 for the course PHY 59372 taught by Professor Fitzpatrick during the Spring '10 term at University of Texas at Austin.

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7hw7 - Physics 315: Oscillations and Waves Homework 7: Due...

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