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Laplace Transforms, Region of Convergence

  1. Oct 26, 2009 #1
    Can anyone explain the region of convergence to me in english? I understand the Laplace transform and can do it with my eyes closed, but I cant figure out how to figure out the ROC. Ive scoured the internet, and every definition is vague or just incomprehensible by me.


    Thanks!
     
  2. jcsd
  3. Oct 26, 2009 #2

    lurflurf

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    Homework Helper

    Do you know anything about complex variables? The region of convergence is just the values of t where
    [tex]\int_0^\infty f(t)e^{-s t} dt[/tex]
    converges as an improper integral.
    That can be difficult to find in general, but in many elementary applications only very well behaved f are considered for example the functions of exponential order.
     
  4. Oct 27, 2009 #3
    I'm somewhat familiar with complex variables, although not too much. I guess what im not really sure of, is what exactly converges? The function and e^-st?


    Thanks for your reply!
     
  5. Oct 27, 2009 #4
    What exactly converges? The improper integral. That is, the limit
    [tex]\lim_{M\to+\infty}\int_0^M f(t)e^{-s t} dt[/tex]
    exists. Generally, the region of convergence is a half-plane: all [itex]s[/itex] to the right of some vertical line in the complex plane.
     
  6. Oct 29, 2009 #5

    jasonRF

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    Gold Member

    As an example to what g_edgar wrote, consider
    [tex] f(t) = e^{5 t}, [/tex]
    for
    [tex]t\geq0. [/tex]

    Now calculuate

    [tex]
    \lim_{M\to+\infty}\int_0^M f(t)e^{-s t} dt = \lim_{M\to+\infty}\int_0^M e^{-(s-5) t}.
    [/tex]

    You should find that the limit only converges if [tex]s[/tex] satisfies some condition. That condition defines the region of convergence. Note that [tex]s[/tex] is complex in general, and the constraint will be on the real part.

    Jason
     
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