How to find the Laplace transformation

Question

Given $\mathscr{L}\left\{\cos(8t)/\sqrt{\pi t}\right\}=e^{-8/s}/\sqrt{s}$

How to find the Laplace transformation

$\mathscr{L}\{\cos(8t)\sqrt{t/\pi}\}$?

Answer 1

$$\cal L\left\{\frac{ \cos 4\sqrt{2 t}} {\sqrt{\pi t}} u(t) \right\} = \frac{e^{-8/s}}{\sqrt{s}}. $$

Since $$\frac{ \sqrt{t} \cos 4\sqrt{2 t}} {\sqrt{\pi }} =\frac{ \cos 4\sqrt{2 t}} {\sqrt{\pi t}} \cdot t,$$

Integrating-by-parts $\Longrightarrow$ Laplace transform of $t \cdot f(t) $ is $- \cal L \left\{f(t)\right\}$.

We have:

$$\cal L\left\{\frac{ \sqrt{t} \cos 4\sqrt{2 t}} {\sqrt{\pi }} u(t) \right\} =-\frac{d}{ds} \cal L\left\{\frac{ \cos 4\sqrt{2 t}} {\sqrt{\pi t}}u(t) \right\} =-\frac{d}{ds} \frac{e^{-8/s}}{\sqrt{s}} =\frac{e^{-8/s}(s-16)}{2s^{5/2}}.$$