ii. Prove the second translation theorem (in time): If F(s) = L(f(t)), then L(ua(t)f(t – a)) = e-a® F(s) (a > 0).
ii. Prove the second translation theorem (in time): If F(s) = L(f(t)), then L(ua(t)f(t – a)) = e-a® F(s) (a > 0).
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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Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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Question
Part ii)
![5. Let e be a positive number and consider the function f.(x) defined by
if 0 <x< €
f.(x)=-
%3D
if x > €.
The graph of this function is shown in Figure 65. It is clear that for
every e > 0 we have Jo f.(x)dx = 1. Show that
1-e-pe
L[f.(x)] =-
pe
and
lim L[f.(x)]=1.
Strictly speaking, lim0 f.(x) does not exist as a function, so
L[lim.0 f.(x)]is not defined; but if we throw caution to the winds, then
8(x) = lim f.(x)
1/ɛ
FIGURE 65](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe25f92a2-1c4d-41a3-af19-0cdf00d27604%2Fd724cf5e-af93-44ae-9e54-721a3fadd1cf%2Fv93m9l8_processed.png&w=3840&q=75)
Transcribed Image Text:5. Let e be a positive number and consider the function f.(x) defined by
if 0 <x< €
f.(x)=-
%3D
if x > €.
The graph of this function is shown in Figure 65. It is clear that for
every e > 0 we have Jo f.(x)dx = 1. Show that
1-e-pe
L[f.(x)] =-
pe
and
lim L[f.(x)]=1.
Strictly speaking, lim0 f.(x) does not exist as a function, so
L[lim.0 f.(x)]is not defined; but if we throw caution to the winds, then
8(x) = lim f.(x)
1/ɛ
FIGURE 65
Transcribed Image Text:ii. Prove the second translation theorem (in time): If F(s) = L(f (t)), then
L(ua(t)f(t – a)) = e_a$ F(s) (a > 0).
-as
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