Laplace transform problem

Question

This is the problem. I know ℒ (t^n) = ℒ (t^n-1) x n/s, which should give us -2/sqrt(s). That's as far as I've come and I don't get the hint either. Where does the pi even come from?

Answer 1

Use that, when $\Re[s]>0$:

$$\mathcal{L}_t\left[t^n\right]_{(s)}=\int_{0}^{\infty}t^ne^{-st}\space\text{d}t=\frac{s^{n+1}}{\Gamma(1+n)}=\frac{s^{n+1}}{n!}$$

$$\mathcal{L}_t\left[\left(4t\right)^{-\frac{1}{2}}\right]_{(s)}=\mathcal{L}_t\left[\frac{1}{\left(4t\right)^{\frac{1}{2}}}\right]_{(s)}=\mathcal{L}_t\left[\frac{1}{2\sqrt{t}}\right]_{(s)}=$$ $$\frac{1}{2}\mathcal{L}_t\left[\frac{1}{\sqrt{t}}\right]_{(s)}=\frac{1}{2}\mathcal{L}_t\left[t^{-\frac{1}{2}}\right]_{(s)}=\frac{1}{2}\cdot\frac{s^{-\frac{1}{2}+1}}{\left(-\frac{1}{2}\right)!}=\frac{\sqrt{\pi}}{2\sqrt{s}}$$

And we know that: $\left(-\frac{1}{2}\right)!=2\left(\frac{1}{2}\right)!=\sqrt{\pi}$