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

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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 σ 2 x parenrightbigg . Demonstrate that ¯ F ( k ) ≡ 1 2 π integraldisplay ∞ −∞ F ( x ) e i k x dx = 1 radicalbig 2 π σ 2 k exp parenleftbigg- k 2 2 σ 2 k parenrightbigg , where i is the square-root of minus one, and σ k = 1 /σ 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 = √ π .] Hence, show from de Moive’s theorem, exp( i θ ) ≡ cos θ + i sin θ , that C ( k ) ≡ 1 2 π integraldisplay ∞ −∞ F ( x ) cos( k x ) dx = 1 radicalbig 2 π σ 2 k exp parenleftbigg- k 2 2 σ 2 k parenrightbigg , S ( k ) ≡ 1 2 π integraldisplay ∞ −∞ F ( x ) sin( k x ) dx = 0 . 2. Demonstrate that integraldisplay ∞ −∞ 1 radicalbig 2 π σ 2 k exp parenleftbigg- k 2 2 σ 2 k parenrightbigg dk...
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7hw7 - Physics 315 Oscillations and Waves Homework 7 Due in...

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