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Laplace Transform for t^n.

  1. Feb 18, 2006 #1
    Our professor asked us to derive an expression for the laplace transfrom of t^n. I did a few examples in MatLab and gathered that the Laplace Transform of t^n = n!/s^(n+1). I'm pretty sure this is correct, but I don't think my professor will be happy with it. I don't really know how I should go about proving it in a more sturdy way. I know I can integrate by parts for specific examples, but I'm not versed in integrating by parts with n's.

    Any Suggestions?
     
  2. jcsd
  3. Feb 18, 2006 #2

    FredGarvin

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    Science Advisor

    I don't see how you can get away without doing integration by parts. What is the definition of the Laplace transform?

    [tex]
    L \{f(t) \} = \int_{0}^{\infty} f(t) e^{-st} dt
    [/tex]
    BTW, you know that the n's are constants, right?
     
    Last edited: Feb 18, 2006
  4. Feb 18, 2006 #3
    Let's say if you have

    [tex]x(t) \iff X(s)[/tex]

    and you wish to find the Laplace transform of

    [tex]t x(t)[/tex]

    Differentiate with respect to s of the Laplace transform integral. That is

    [tex]\frac {d}{ds} \int_{0^-}^\infty t x(t) e^{-st} dt[/tex]

    You may move the derivative inside the integral and differentiate the exponential of the integrand.

    Doing so you will see that [tex]t x(t) \iff - \frac {dX(s)}{s}[/tex]

    Try generalizing this for [itex]t^n[/itex]. Note that for your specific problem

    [tex] x(t) = t [/tex]
     
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