1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Another Laplace Transform problem, need region of convergence help

  1. Nov 15, 2014 #1
    1. The problem statement, all variables and given/known data

    Find L[x(t)], where $$ x(t) = tu(t) + 3e^{-1}u(-t) $$

    Also determine the region of convergence

    2. Relevant equations

    Laplace properties, Laplace table:

    L[te-at = 1/(s+a)2

    L[u(t)] = 1/s

    L[t] = 1/s2

    3. The attempt at a solution

    I don't really know what to do with this as my table doesn't give the product of these two.

    Do you just combine them like this?:

    tu(t) ---> (1/s)(1/s2)

    3e-3u(-t) ---> wait...

    I don't even know if my table says anything about a transform for u(-t), could it be -1/s?

    I was going to do this and then add them to each other. Also need help with RoC
  2. jcsd
  3. Nov 15, 2014 #2


    User Avatar

    Staff: Mentor

    No. You make use of this rule:

    L t.f(t) = -F'(s)

    so if you have a function f(t) whose Laplace transform you know to be F(s)
    then multiplying that function by t results in a Laplace transform which can be calculated by you as the derivative of F(s) multiplied by -1

    F'(s) means the derivative of F(s). So you need to know how to differentiate F(s) with respect to s.
    Last edited: Nov 15, 2014
  4. Nov 15, 2014 #3


    User Avatar

    Staff: Mentor

    Check that e-1
    If it's really e-1 then that's just a constant, it's a number.

    I don't know anything about u(-t)

    Paging rude man ☎ ....
  5. Nov 16, 2014 #4
    Yeah that was actually -t and I read it too fast.

    So since I'm converting it to s-domain does that mean I integrate it instead (so it becomes F(s) now)?
  6. Nov 16, 2014 #5


    User Avatar

    Staff: Mentor

    No. You start with F(s) and by differentiating F(s) you're calculating F'(s) which is what you need.
  7. Nov 16, 2014 #6


    User Avatar

    Staff: Mentor

  8. Nov 16, 2014 #7
    Ok for that the derivative is tδ(t) + u(t)? And that's it at least for that right? Still not sure what RoC is
  9. Nov 16, 2014 #8


    User Avatar

    Staff: Mentor

    We're looking at only the t.u(t) term right now.

    What is the Laplace Transform of u(t)?
    Last edited: Nov 16, 2014
  10. Nov 16, 2014 #9

    rude man

    User Avatar
    Homework Helper
    Gold Member

    This is a math problem more than an engineering problem. The usual application of the Laplace transform is to solve a linear differential equation with constant coefficients and with given initial conditions.

    This problem on the other hand is purely math and probably purely useless, but here goes:

    The mathematically correct Laplace transform is L{f(t)} = integral from -∞ to +∞ of f(t)exp(-st)dt.
    In the real problems to which I referred, the transform is integral from 0 to +∞ of the same integrand.
    Thus, taking your expression, the fact that it includes a u(-t) forces you to integrate from -∞ rather than zero.
    In other words, and we've gone thru all this before, pick you limits to accord with the u function's argument.

    As to convergence, look at the given time function. The region of convergence is simply the region where the time function does not blow up to ∞. The "region" is the region in the complex s plane, with x axis = σ = Re{s} and y axis = Im{s} = jw.

    f(t) = exp(-at) u(t)
    F(s) = integral fro 0 to infinity of exp(-at)exp(-st)dt
    = integral from 0 to infinity of exp[-(s+a)t]dt
    = (-1/(s+a)[exp(-(s+a)t evaluated from t=0 to infinity.

    Now, you can see that, since 0<t<∞, the expression (s+a) must be positive or you get infinity for evaluating the integral between its limits.
    But s = σ + jw
    So σ must be > -a
    and the region of convergence is the region to the right of σ = -a in the s plane, sine there σ> -a.

    The u(t) term is handled similarly.

    Look at the attached for more info.

    Attached Files:

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Another Laplace Transform problem, need region of convergence help
  1. Laplace Transform help (Replies: 1)