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pset_4b - ACM 95/100c Problem Set 4 16th April 2006 Due by...

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Unformatted text preview: ACM 95/100c Problem Set 4 16th April 2006 Due by 5:00PM on 4/29/2006 Please deposit all problem sets in the slot in 303 Firestone Please remember to include your section number and section instructor 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 δ-function but also interprets the δ-function as an extension of the Kro- necker δ that appears when we examine the orthogonality of Sturm-Liouville eigenfunctions on nite domains (like sines and cosines). (a) For what α does the function α exp(- β ( x- x ) 2 ) have unit area for-∞ < x < ∞ ? (b) Show that in the limit as β → ∞ ,the resulting function in part (a) satis es the properties of the Dirac delta function δ ( x- x ) . (c) Obtain the Fourier transform of δ ( x- x ) in two ways: i. Take the transform of part (a) and take the limit as β → ∞ ....
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