For any function ft the laplace transform is defined

Info icon This preview shows pages 17–29. Sign up to view the full content.

View Full Document Right Arrow Icon
For any function f(t), the Laplace Transform is defined as 1.1 Introduction to Laplace Transforms ( 29 0 ( ) st f t e f t dt - = So, how exactly is the Laplace Transform defined?
Image of page 17

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
As this is a definite integral between limits on t, the final result will NOT depend on t. It will, however depend on s. For this reason, [f(t)] is often written as 1.1 Introduction to Laplace Transforms ( 29 0 ( ) st f t e f t dt - = ( ) f s
Image of page 18
Example 1.1.1 If f(t) = t, find the Laplace Transform ( 29 [ ] ( ) f t t f s = = 1.1 Introduction to Laplace Transforms Well, in this case, this appears to be a simple integration. Do you remember how to do it ? - = 0 ] [ dt te t st
Image of page 19

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
( 29 [ ] at f t e = Example 1.1.2 If , find the Laplace Transform Hence, for example with , find 1.1 Introduction to Laplace Transforms ( ) at f t e = - = 0 4 4 ] [ dt e e e st t t 4 = a
Image of page 20
Laplace Transforms : An Illustration Consider the function f(t) = t e -st for different values of s (s = 1, 2, 5). 1.1 Introduction to Laplace Transforms
Image of page 21

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
is given by the area under the curve for the appropriate value of s. 1.1 Introduction to Laplace Transforms [ ] 0 st t t e dt - =
Image of page 22
s = 1 : Largish area under curve 1.1 Introduction to Laplace Transforms
Image of page 23

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
s = 2 : Smaller area under curve 1.1 Introduction to Laplace Transforms
Image of page 24
s = 5 : Even smaller area under curve 1.1 Introduction to Laplace Transforms
Image of page 25

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
The Laplace Transform (area under curve) depends on s. In fact, it is equal to 1/s 2 1.1 Introduction to Laplace Transforms
Image of page 26
1.2 Laplace Transforms by Table
Image of page 27

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
The direct integration approach of section 1.1 will give the Laplace Transform of many functions. Once carried out, these Laplace Transforms can be written in a table.
Image of page 28
Image of page 29
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern