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(d) Show that the Fourier transform of a Gaussian is a Gaussian, and hence show that eik dk 2πδ(x).

(d) Show that the Fourier transform of a Gaussian is a Gaussian, and hence show that eik dk 2πδ(x).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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plz solve question (d) with explanation get multiple upvotes
1. A Pleasant Evening with Delta "functions" Let's define the Dirac delta "function" 8(x) by the property | f(x)8(x) dx = f (0), for "well-behaved" functions f. (a) Consider the family of box functions 1 2a -a < x < a, Ba(x) else. Show that in the a → 0 limit, Ba goes to the delta function. (Hint: Taylor expand f around 0) (b) Show that the Gaussian with u (Hint: use a change of variables to show that (x") = Cno", for some constant Cm which you need to evaluate only for n 0 goes to the delta function as o → 0. (c) What are the values of 8(x) for x # 0, x = 0? Is it a well-defined function? (d) Show that the Fourier transform of a Gaussian is a Gaussian, and hence show that oika dk 2π (π). | e
Transcribed Image Text:1. A Pleasant Evening with Delta "functions" Let's define the Dirac delta "function" 8(x) by the property | f(x)8(x) dx = f (0), for "well-behaved" functions f. (a) Consider the family of box functions 1 2a -a < x < a, Ba(x) else. Show that in the a → 0 limit, Ba goes to the delta function. (Hint: Taylor expand f around 0) (b) Show that the Gaussian with u (Hint: use a change of variables to show that (x") = Cno", for some constant Cm which you need to evaluate only for n 0 goes to the delta function as o → 0. (c) What are the values of 8(x) for x # 0, x = 0? Is it a well-defined function? (d) Show that the Fourier transform of a Gaussian is a Gaussian, and hence show that oika dk 2π (π). | e
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