ps4sol - ACM 95/100c Problem Set 4 Solutions Lei Zhang...

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Unformatted text preview: ACM 95/100c Problem Set 4 Solutions Lei Zhang April 22, 2006 Each problem is worth 10 points - each part of a multi-part problem is weighted equally Problem 1 (Collaboration allowed) The following problem examines the Fourier transform for the -function but also interprets the -function as an extension of the Kronecker that appears when we examine the orthogonality of Sturm-Liouville eigenfunctions on finite domains (like sines and cosines). (a) For what does the function exp(- ( x- x ) 2 ) have unit area for- < x < ? (b) Show that in the limit as ,the resulting function in part (a) satisfies the properties of the Dirac delta function ( x- x ). (c) Obtain the Fourier transform of ( x- x ) in two ways: i. Take the transform of part (a) and take the limit as . ii. Use the integration properties of the Dirac delta function. (d) Show that the transform of ( x- x ) is consistent with the following idea: Transforms of sharply peaked functions are spread out (that is they contain lots of frequencies). (e) Interpret the following equation ( - ) = 1 2 Z - exp(- ix ( - )) dx. as a type of orthogonality relationship for the eigenfunctions exp(- ix ). (f) Use this orthogonality relationship to compute F ( ) if f ( x ) = Z - F ( )exp(- ix ) d. This is a way of computing the Fourier coefficients of f ( x ) otherwise known as the Fourier transform of f ( x ) using the appropriate orthogonality condition and the appropriate eigenfunctions. Solution 1 (a) For the function to have unit area for- < x < means that 1 Z - exp - ( x- x ) 2 / dx = Z - exp (- y 2 ) dy = Z - exp (- z 2 ) dz = = 1 . (1) So, = r . (2) (b) Let f ( x ) r exp - ( x- x ) 2 / . (3) For x 6 = x , lim f ( x ) = 0 , (4) since exp - ( x- x ) 2 / decays in faster than grows. Also, f ( x ) = q becomes unbounded as . Finally, by part a), Z - f ( x ) dx = 1 (5) for all . Thus, as , f ( x ) satisfies the properties of ( x- x ). 2 (c) i. 1 2 Z - r exp - ( x- x ) 2 / exp( ix ) dx = r exp (- x 2 ) 1 2 Z - exp - x 2- 2 xx- i x dx = r exp (- x 2 ) 1 2 Z - exp "- x- x- i 2 2 + x + i 2 2 # dx = r exp- x 2 + x 2 + ix- 2 4 1 2 Z - exp "- x- x- i 2 2 # dx = 1 2 exp ix- 2 4 r Z - exp (- y 2 ) dy = 1 2 exp ix- 2 4 1 2 exp( ix ) as . (6) ii. By the properties of the Dirac function, 1 2 Z - ( x- x )exp( ix ) dx = 1 2 exp( ix ) . (7) Note (not necessary for students to state): It requires complex analysis to justify Z - exp "- x- x + i 2 2 # dx = Z - exp (- y 2 ) dy. (8) This justification is not required here, though. See Habermans Appendix to 10.3 for details.This justification is not required here, though....
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ps4sol - ACM 95/100c Problem Set 4 Solutions Lei Zhang...

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