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Looking for a proof

  1. Sep 11, 2005 #1
    Hey guys, how can this Laplace transform be proven. I always see it in the tables but don't know how it came to be.

    [tex]L[t^n] = \frac{n!}{s^{n+1}}[/tex]
     
  2. jcsd
  3. Sep 11, 2005 #2
    Just do the transformation. With a simple change of variables you can reduce the integral to the gamma function.
     
  4. Sep 11, 2005 #3
    Ah. Ok I got it thanks. I guess I must have let it slip by me that gamma function is related.
     
  5. Sep 11, 2005 #4
    here is a formal proof:
    You can show that
    [tex] \int t^n e^{-ts} dt = -s^{-n-1}\int_{st}^{\infty}x^n e^{-x}dx +c [/tex]
    therefore
    [tex] \int_{0}^{\infty} t^n e^{-ts} dt =s^{-n-1}\int_{0}^{\infty}x^n e^{-x}dx =s^{-n-1} n![/tex]
     
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