Problem 2.24 Delta functions live under integral signs, and two expressions (D₁(x) and D₂(x)) involving delta functions are said to be equal if +∞ +00 [ f(x)D₁(x) dx = [ f(x)D₂(x) dx. for every (ordinary) function f(x). (a) Show that 8(cx) = -8(x) |c| [2.142] where c is a real constant. (Be sure to check the case where c is negative.) (b) Let (x) be the step function: 1. if x > 0. 0. if x < 0. 0(x)= [2.143] (In the rare case where it actually matters, we define (0) to be 1/2.) Show that de/dx = 8(x).
Problem 2.24 Delta functions live under integral signs, and two expressions (D₁(x) and D₂(x)) involving delta functions are said to be equal if +∞ +00 [ f(x)D₁(x) dx = [ f(x)D₂(x) dx. for every (ordinary) function f(x). (a) Show that 8(cx) = -8(x) |c| [2.142] where c is a real constant. (Be sure to check the case where c is negative.) (b) Let (x) be the step function: 1. if x > 0. 0. if x < 0. 0(x)= [2.143] (In the rare case where it actually matters, we define (0) to be 1/2.) Show that de/dx = 8(x).
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![Problem 2.24 Delta functions live under integral signs, and two expressions (D₁ (x)
and D₂(x)) involving delta functions are said to be equal if
[ + f(x)D₁(x) dx = [+0° f(x)D₂(x) dx.
-8
-00
for every (ordinary) function f(x).
(a) Show that
8(cx) =
-178(x).
[2.142]
where c is a real constant. (Be sure to check the case where c is negative.)
(b) Let 0(x) be the step function:
{
0(x)=
1.
[2.143]
(In the rare case where it actually matters, we define (0) to be 1/2.) Show
that de/dx = 8(x).
if x > 0.
if x < 0.
0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F668ff5f2-d557-46a3-9b4c-7a2704e0b63a%2F06ecc679-1fc1-4ae1-b53e-b3c29106baa3%2F1kcidgm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 2.24 Delta functions live under integral signs, and two expressions (D₁ (x)
and D₂(x)) involving delta functions are said to be equal if
[ + f(x)D₁(x) dx = [+0° f(x)D₂(x) dx.
-8
-00
for every (ordinary) function f(x).
(a) Show that
8(cx) =
-178(x).
[2.142]
where c is a real constant. (Be sure to check the case where c is negative.)
(b) Let 0(x) be the step function:
{
0(x)=
1.
[2.143]
(In the rare case where it actually matters, we define (0) to be 1/2.) Show
that de/dx = 8(x).
if x > 0.
if x < 0.
0.
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