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