Laplace Transforms
In the last few chapters, we have looked at several ways to use
The Laplace transform is defined in terms of an integral as
7
e""
ft
Note that the input to a Laplace transform is a function of time, /(/), and the output is a function of frequency, F(j), Although many real-world examples require the use of
Let's stan with a simple example. Here we calculate the Laplace transform of /(f) = t. We have
This is an improper integral, so we express it in terms of a limit, which gives
Now we use integration by pans to evaluate the integral. Note that we are integrating with respect to t, so we treat the variable s as a constant. We have
u—t dv — dt
du=dt v
— — ye_ir.
Then we obtain
= + +
= ~K + °1 -
= JinL[[-i,-]-±[e--lj]
- c + c
= 0-0 + -L
s“
_x
2* s
1. Calculate the Laplace transform of /(/) = 1.
3.Calculate the Laplace transform of /(/) = : (Note, you will have to integrate by parts twice.)
Laplace transforms are often used to solve differential equations. Differential equations are not covered in detail until later in this book; but, for now, let’s look at the relationship between the Laplace transform of a function and the Laplace transform of its derivative.
Let’s start with the definition of the Laplace transform. We have
WW! = r™ r'
= lim / e~st fifth.
4.Use integration by parts to evaluate Jjm^ e~sl fifth. (Let « = /{/) and dv — e '!dt.) After integrating by parts and evaluating the limit, you should see that
Then,
Thus, differentiation in the time domain simplifies to multiplication by s in the frequency domain.
The final thing we look at in this project is how the Laplace transforms of fit] and its antiderivative are
related. Let g(r) — f(u}dii. Then,
¦'o
lim /
;-* caj" 5.Use integration by parts to evaluate hrn^y e ’ g(t)dl. (Let u = gif) and dv = e dt. Note, by the way,
that we have defined gif, du — fifth.)
As you might expect, you should see that
L|^(r)| = |-L[/(/)i.
Integration in the time domain simplifies to division by s in ±e frequency domain.
Want to see the full answer?
Check out a sample textbook solutionChapter 3 Solutions
Calculus Volume 2
Additional Math Textbook Solutions
Calculus Volume 1
Introductory Statistics
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
Mathematics for Elementary Teachers with Activities (5th Edition)
Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
Probability and Statistics for Engineers and Scientists
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage