# Math Help - Laplace Transform Gamma Function

1. ## Laplace Transform Gamma Function

I am just looking for a proof or help to solve the Laplace Transform that:

f(t^α) = [ Γ(α+1) ] / [ s^(α+1) ]

I dont even know where to begin working on this and any help would be greatly appreciated.
Thanks.

2. You need the definition of the Laplace transform and make a substitution as follows:
$\int_0^{+\infty}e^{-st}t^a dt$
Now set:
$st=u \qquad s>0$
You can rewrite this after the substitution as:
$\frac{1}{s^{a+1}}\int_0^{+\infty}e^{-u}u^a du$
Which can be written using the definition of the Gamma function to the required result.

3. I'm not seeing how substituting in u for st is turning the t into a u.

4. Originally Posted by eeverett1
I'm not seeing how substituting in u for st is turning the t into a u.
$\int_0^{\infty} e^{-st}t^a dt = \int_0^{\infty} e^{-st} \left( \frac{st}{s} \right)^a dt = \frac{1}{s^a}\int_0^{\infty} e^{-st}(st)^a dt$.
Now let $\mu = st$ so then $\frac{1}{s^{a+1}} \int_0^{\infty} e^{-\mu} \mu^a d\mu$.