laplace transform multiplication by power t

Question

How to solve Laplace transform of $\displaystyle t^\frac{5}{2} e^{4t}$ . I know that this can be solve by multiplication by power of $t$ but how to differentiate $\frac{5}{2}$ part please reply

Answer 1

For non-integer powers of $t$ it might be easier to solve without differentiating:

It holds that (if $\text{Re}\,a>-1$ and $\text{Re}\,s>0$) $$ \int_0^{+\infty}t^a e^{-st}\,dt = \frac{\Gamma(a+1)}{s^{a+1}}, $$ so $$ \mathcal{L}(t^{5/2})=\frac{\Gamma(7/2)}{s^{7/2}}=\frac{15\sqrt{\pi}}{8s^{7/2}}. $$ The function $e^{4t}$ just shifts this, so (for $\text{Re}\,s>4$) $$ \mathcal{L}(t^{5/2}e^{4t})=\frac{15\sqrt{\pi}}{8(s-4)^{7/2}}. $$