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Limit properties laplace transform

  1. Aug 1, 2011 #1

    vrc

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    hello,

    I'am stuying the laplace transform and have problems with understanding the follow thing my course says:

    lim s->+inf F(s)=lim t->0 f(t)
    lim s->0 F(s)=lim t->+inf. f(t)

    I would like to understand it because I have the feeling that this are important properties.

    Thank you very much !
     
  2. jcsd
  3. Aug 1, 2011 #2
    OK, maybe I'm making a dumb mistake, but I don't think this is true. For instance, let [itex]f(t) = 1[/itex]. The Laplace Transform of f is [itex]F(s) = \frac{1}{s}[/itex]. [itex]lim_{s\to\infty}F(s) = 0[/itex]. [itex]lim_{t\to 0}f(t) = 1[/itex].

    Could it be that there are some other conditions on these limit theorems that you forgot to mention?
     
  4. Aug 1, 2011 #3

    vrc

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    this is what's written in my course:

    first the derivation (not mathematical derivation!) of the laplace transform of a periodic funtion f
    than comes those properties, the source where it comes from says: form mechanical point of view this are limit properties of the laplace transform form a function f(t)

    there are no specific assumptions for those equations written but indeed it doens't make sense...

    thank you
     
  5. Aug 1, 2011 #4
    Ah! Well, that's at least one condition you didn't mention. You didn't say in the OP that f was periodic.

    But it still doesn't seem to work. For instance, the Laplace transform of cos(t) is s/(1+s2). This goes to 0 as s->infinity, but cos(0) = 1.
     
  6. Aug 1, 2011 #5

    vrc

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    no no,

    it's about a function f(t) but never explicit formulated that this is a periodic function...
    de derivation of de laplace transform of a periodic function is just on the same space
    have not the idea of a link between them.

    The only thing it says is that from mechanical side those limit properties are derived...
    don't understand it..

    htank you
     
  7. Aug 1, 2011 #6
    What does "from mechanical side" mean?
     
  8. Aug 1, 2011 #7
    I think you're talking about the initial and final value theorems. If this is the case, you've written them down wrong. They state:

    IVT: f(0+) = lim(s->Infinity) s*F(s)
    FVT: f(Infinity) = lim(s->0) s*F(s)

    These theorems only work if the poles of the Laplace transform are on the left hand side of the y-axis. They are helpful in mechanical system dynamics because you don't have to do the partial fraction decomposition in order to find out what the system will do in the long term.
     
  9. Aug 2, 2011 #8

    vrc

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    I suppose , Idon't have to worry about I don't understand it because I didn't examinate the mechanical system dynamics linked to laplace transform yet.

    Anyway thank you vm !
     
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