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Laplace transform ?

  1. Jul 15, 2009 #1
    1. The problem statement, all variables and given/known data
    t*e^(it) how do we take the laplace transform of this .

    would it be 1/((s-i)^2) then how would we get the real part of that .
     
  2. jcsd
  3. Jul 15, 2009 #2
    I'm not quite sure how the Laplace transform of a complex-valued function such as e^(it) works, but sense you want to take the real part of that is it the case that what you're really looking for the Laplace transform of is t*cos(t)? Because that isn't so hard and introducing complex numbers seems like the long way around.
     
  4. Jul 15, 2009 #3
    then how would i take the laplace transform of t*cos(t) based on if i knew the
    lapace tranform of t and cos(t) how does this product work out.
     
  5. Jul 15, 2009 #4
    A handy result in the theory of Laplace transforms says that:
    [tex]L\{t^nf(t)\}(s)=(-1)^n\frac{d^n}{ds^n}L\{f\}(s)[/tex]
    which we can use to compute:
    [tex]L\{t\cos{t}\}=-\frac{d}{ds}L\{\cos{t}\}[/tex]
    Now all you need to do is look up the transform for [itex]\cos{t}[/itex], differentiate and you're on your way home!
     
  6. Jul 15, 2009 #5
    thanks , what is this rule called .
     
  7. Jul 15, 2009 #6
    That's a good question; I'm not sure if this identity has a name or not. Perhaps someone else will know.

    It's not too hard too prove. All you do is write out the Laplace transform for t^n * f(t) and use integration by parts n-times (which you could also do to get the answer with any ol' f(t), but if you remember it, the identity is pretty quick).
     
  8. Jul 16, 2009 #7
    the only problem is my computer wont let me see the latex black boxes so i am having to look at it how you typed it in so i am having trouble reading it .
     
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