Laplace Transformation of Negative exponent Real Numbers like $t^\frac{-3}{2}$

Question

How would we apply LT for functions that has exponents as negative real numbers. I am not sure how the highlighted portion in below solution is arrived at.

Answer 1

$\int_0^\infty t^{a-1} e^{-st}dt$ diverges whenever $\Re(a) \le 0$. So you need to regularize it for making sense to $\mathcal{L}\{t^{-3/2}\}$, for example : $$F_a(s) = \int_0^\infty t^{a-1} (e^{-st}-e^{-t})dt$$ One can show that (change of variable $x = st$) $$F_a(s) = \int_0^\infty (x/s)^{a-1} e^{-x}\frac{dx}{s}-\int_0^\infty t^{a-1}e^{-t}dt=(s^{-a} -1)\Gamma(a)$$ for $\Re(a) > 0, s > 0$ and by analytic continuation this stays true for $\Re(a) > -1, \Re(s) > 0$.