Looking for a proof

  • Thread starter mathwurkz
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  • #1
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Main Question or Discussion Point

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]
 

Answers and Replies

  • #2
574
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Just do the transformation. With a simple change of variables you can reduce the integral to the gamma function.
 
  • #3
41
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Ah. Ok I got it thanks. I guess I must have let it slip by me that gamma function is related.
 
  • #4
2
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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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