1. Show that the Laplace transform of f is F(s) = r- Fi(s). Hint: Split the domain [0, 0] as [0, T]U [T, ∞0]; for the second integral, do a change of variable to make the integral look like the standard Laplace integral. %3D 1–e-

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 51E
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III) Let function ƒ be periodic with period T > 0, that is, f(t) = f(t+T), for -∞ <t <
o. Define
T
L =*f(t)dt.
-st
F;(s) =
0,
1. Show that the Laplace transform of f is F(s)
Hint: Split the domain [0, 0] as [0, T]U [T, ∞]; for the second integral, do a change
of variable to make the integral look like the standard Laplace integral.
1
F:(s).
1-e-sT
2. Use the result to compute the Laplace transform of the half-wave rectified sine signal
given by,
sin wt
2nt
<t<
(2n+1)r
f(t) =
%3D
(2n+1)r < t <
(2n+2)т
n = 0, 1, 2, ...
You may use:
| eat sin(wt)dt =
eat
(-w cos(wt) + a sin(wt).
a² + w²
Transcribed Image Text:III) Let function ƒ be periodic with period T > 0, that is, f(t) = f(t+T), for -∞ <t < o. Define T L =*f(t)dt. -st F;(s) = 0, 1. Show that the Laplace transform of f is F(s) Hint: Split the domain [0, 0] as [0, T]U [T, ∞]; for the second integral, do a change of variable to make the integral look like the standard Laplace integral. 1 F:(s). 1-e-sT 2. Use the result to compute the Laplace transform of the half-wave rectified sine signal given by, sin wt 2nt <t< (2n+1)r f(t) = %3D (2n+1)r < t < (2n+2)т n = 0, 1, 2, ... You may use: | eat sin(wt)dt = eat (-w cos(wt) + a sin(wt). a² + w²
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