Quiz6 - Quiz 6. We recall the Fourier transform and inverse...

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Unformatted text preview: Quiz 6. We recall the Fourier transform and inverse Fourier transform formulas: F{ f } ( ) = F ( ) = 1 2 Z - e ix f ( x ) dx, F- 1 { F } ( x ) = f ( x ) = Z - e- ix F ( ) d. 1. For F ( ) = e- | | with > 0, calculate the inverse Fourier transform f ( x ). 2. Show that the Fourier transform of f ( x ) is- iF ( ) and that the Fourier transform of xf ( x ) is- idF/d . 3. What is the Fourier transform of x 2 f ( x )?, the Fourier transform of f 00 ( x )? 4. Calculate the Fourier transform of f ( x- ) for some constant . Calculate the Fourier transform of f ( x ) = 2 ( x- 1) 2 + 1 . Solutions. 1. For > 0 we have Z e- ix e- d = 1- ix- e- ix- = 1- ix- (0- 1) = 1 ix + , and for < 0, Z- e- ix e + d = 1- ix + e- ix + - = 1- ix + (1- 0) = 1- ix + . In both cases, this is because > 0. Therefore, f ( x ) = 1 ix + + 1- ix + = 2 ( ix + )(- ix + ) = 2 x 2 +...
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This note was uploaded on 02/05/2012 for the course APMAE 4200 taught by Professor R during the Fall '11 term at Columbia.

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