Laplace Transform of e^(-t)cos2t u(t-1)

Click For Summary

Homework Help Overview

The discussion revolves around finding the Laplace transform of the function e^(-t)cos(2t) multiplied by the unit step function u(t-1). Participants are exploring the implications of the step function on the limits of integration in the context of the Laplace transform.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss changing the limits of integration due to the presence of the step function, with some questioning whether this affects the function itself. There are references to trigonometric identities and derivative rules as potential tools for evaluating the integral.

Discussion Status

The discussion is active, with participants confirming their understanding of the limits of integration and exploring different aspects of the integral. There is no explicit consensus yet, but various interpretations and approaches are being considered.

Contextual Notes

Participants are navigating the implications of the unit step function on the integration process, particularly regarding the transformation of the function itself when adjusting limits. There is uncertainty about whether the changes in limits necessitate modifications to the variables within the function.

Moneer81
Messages
158
Reaction score
2

Homework Statement



What is the laplace transform of e^(-t) cos 2t u(t-1)


Homework Equations



definition of Laplace transform: LT of f(t) = integral of f(t)e^-st dt, where limits of integration are from 0 to infinity


The Attempt at a Solution



since I have u(t-1) then do I just change the limits of integration to go from 1 to infinity instead?


then I guess what is the fastest way to evaluate the resulting integral:
integral of e^-(s+1)t cos 2t dt ?
 
Physics news on Phys.org
my hint is that the step function has a big influence on the limits of integration since the step function is zero to the left side of when the step function is one.

and then...maybe I am not sure, you could use a trigonometry identity
 
OK so I was right in changing the lower limit of integration from 0 to 1?

As far as the integral goes, for a similar problem in a book I was reading they ended up with an expression that was somehow obtained from the quotient rule of derivatives...
 
u(t-1)

take t - 1 = 0 -----> t = 1

but I am not sure if changing the limits of integration forces you to change the "t's" in the function.

ex. cos 2(t-1) and e^-((s+1)(t-1))
 

Similar threads

  • · Replies 1 ·
Replies
1
Views
1K
  • · Replies 10 ·
Replies
10
Views
3K
Replies
2
Views
2K
  • · Replies 4 ·
Replies
4
Views
3K
  • · Replies 3 ·
Replies
3
Views
3K
  • · Replies 6 ·
Replies
6
Views
6K
  • · Replies 4 ·
Replies
4
Views
4K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 1 ·
Replies
1
Views
1K
  • · Replies 2 ·
Replies
2
Views
2K