A Pleasant Evening with Delta “functions" Let's define the Dirac delta "function" 8(x) by the property | f(x)6(x) dr = f(0), for "well-behaved" functions f. (a) Consider the family of box functions -a < x < a, 2a 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 µ = 0 goes to the delta function as o → 0. (Hint: use a change of variables to show that (x") = Cnơ", for some constant Cn which you need to evaluate only for n = 0) (c) What are the values of 8(x) for x # 0, x = 0? Is it a well-defined function?
A Pleasant Evening with Delta “functions" Let's define the Dirac delta "function" 8(x) by the property | f(x)6(x) dr = f(0), for "well-behaved" functions f. (a) Consider the family of box functions -a < x < a, 2a 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 µ = 0 goes to the delta function as o → 0. (Hint: use a change of variables to show that (x") = Cnơ", for some constant Cn which you need to evaluate only for n = 0) (c) What are the values of 8(x) for x # 0, x = 0? Is it a well-defined function?
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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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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