help with laplace transform

Question

Can you please help me with this Laplace transform? I used wolfram alpha to get the answer but I need some hints about the procedure to get to that answer.

$$ \mathcal{L}\left(\frac {\cos(\sqrt{t})}{\sqrt{t}}\right) $$

Answer 1

HINT

write down the integral and set sqrt(t)=u and use substitution method then use integration by parts twice you will get the answer

Answer 2

Hint: You can use Watson's lemma.

Answer 3

Express the Laplace transform as

$$\begin{align} \underbrace{\int_0^{\infty} dt \frac{\cos{\sqrt{t}}}{\sqrt{t}} e^{-s t}}_{t=u^2} &= 2 \int_0^{\infty} du \, \cos{u} \, e^{-s u^2} \\ &= \Re{\int_{-\infty}^{\infty} du \, e^{i u} \, e^{-s u^2}} \\ &= \Re{\int_{-\infty}^{\infty} du \, e^{-s [u^2-i(u/s) - 1/(4 s^2)]} e^{-1/(4 s)}} \\ &= e^{-1/(4 s)} \Re{\int_{-\infty}^{\infty} du \, e^{-s [u-i/(2 s)]^2}} \\ \end{align}$$

As that last integral is $\sqrt{\pi/s}$, the LT sought is

$$\sqrt{\frac{\pi}{s}} e^{-1/(4 s)}$$