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Unformatted text preview: Math 401 (Sec. 502) Spring, 2009 Homework 11 (due April 23) Q1. Find the Fourier transformations of the following functions (a) [Exponentially decaying function] f ( x ) = e- a | x | , a > . (b) [Square wave] f ( x ) = - 1 if- a x 1 if 0 < x a if | x | > a. ( a > 0) (c) [Rectangular pulse function] f ( x ) = braceleftBigg 1 if | x | 1 if | x | > 1 . Q2. (a) If we denote the Fourier transformation F [ f ]( w ) of f ( x ) by f ( w ) then the Fourier inversion formula is given by (assume that f is continuous on the real line) f ( x ) = integraldisplay - f ( w ) e- iwx d w. Use the above inversion formula to prove that f ( x ) = 2 F [ f ](- x ) . (b) Use Q1 (a) and Q2(a) to compute the Fourier transformation of f ( x ) = 1 a 2 + x 2 . Q3. Use Q1 (c) to prove that (a) integraldisplay sin 2 x x 2 d x = 2 [Hint: use Parsevals identity: integraldisplay - | f ( x ) | 2 d x = 2 integraldisplay - | f ( w ) | 2 d w ] (b)...
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- Spring '09