I used the Residue theorem to solve this problem. But, I could not obtain the solution given by
$$\mathscr{L}^{-1}\left( { p^{-3/2}e^{-\sqrt{pa}}\over{2\sqrt{2}}} [\cos(\sqrt{ap})+\sin(\sqrt{ap})] \right)= {\sqrt{a} \over{2}} \left[ S\left({{a}\over{2 t}} \right)-C\left({a}\over{2 t} \right)\right]+{ \sqrt{t \over \pi}}~\sin \left({a \over 2t }+{\pi \over 4} \right) $$
where
$$C(z)={1\over2\pi} \int_{0}^{z} s^{-1/2} \cos s ~ds,$$ and $$S(z)={1\over2\pi} \int_{0}^{z} s^{-1/2} \sin s ~ds.$$
I could find $$ { \sqrt{t \over \pi}}~\sin \left({a \over 2t }+{\pi \over 4} \right)$$ by using integration by parts to inverse Laplace transform ,and in order for Fresnel integrals I need to calculate the following integral
$${2 \sqrt{a}\over \pi} \int_{0}^{\infty} x^{-1} \cos(\sqrt{ax}) \sinh(\sqrt{ax}) e^{-xt} dx. $$
My question is how I can get Fresnel integrals in the solution.
