How can the Laplace transform of L[t^n] be proven?

[mathwurkz]

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.

$$L[t^n] = \frac{n!}{s^{n+1}}$$

 

[inha]

Just do the transformation. With a simple change of variables you can reduce the integral to the gamma function.

 

[mathwurkz]

Ah. Ok I got it thanks. I guess I must have let it slip by me that gamma function is related.

 

[andyL]

here is a formal proof:
You can show that
$$\int t^n e^{-ts} dt = -s^{-n-1}\int_{st}^{\infty}x^n e^{-x}dx +c$$
therefore
$$\int_{0}^{\infty} t^n e^{-ts} dt =s^{-n-1}\int_{0}^{\infty}x^n e^{-x}dx =s^{-n-1} n!$$