How can I solve Laplace Tranformation of $1/s^{5/2}$?

Question

I have just started Laplace Transformation

And I came across a problem which contains $1/s^{5/2}$

How to solve it? I know $\mathcal L\{t^n\}= n!/s^{n+1}$

Please say how to solve it.

Answer 1

Using TravorLZH's idea we have for $x>-1$,

\begin{align*} \mathcal{L}\{t^x\}&=\int_0^\infty t^xe^{-st}dt,\quad\text{doing }u=st\\ &=\int_0^\infty \left(\frac{u}{s}\right)^xe^{-u}\frac{du}{s}\\ &=\frac{1}{s^{x+1}}\int_0^\infty u^xe^{-u}du\\ &=\frac{\Gamma(x+1)}{s^{x+1}} \end{align*}

Taking $x+1=5/2\Rightarrow x=3/2$. We have $$ \mathcal{L}\{t^{3/2}\}=\frac{\Gamma(3/2)}{s^{5/2}}=\frac{\frac{3}{2}\cdot\frac{1}{2}\cdot\Gamma(1/2)}{s^{5/2}}=\frac{3}{4}\cdot\sqrt\pi\cdot\frac{1}{s^{5/2}} $$ Therefore it is easy to conclude that $$ \mathcal{L}^{-1}\left\{\frac{1}{s^{5/2}}\right\}=\frac{4\cdot t^{3/2}}{3\cdot\sqrt\pi}. $$