(a) By making the change of variables $t=\frac{x^{2}}{s}(s > 0)$ in the integral that defines the Laplace transform, show that
$$
L\left[t^{-1 / 2}\right]=2 s^{-1 / 2} \int_{0}^{\infty} e^{-x^{2}} d x
$$
(b) Use your result in (a) to show that
$$
\left(L\left[t^{-1 / 2}\right]\right)^{2}=4 s^{-1} \int_{0}^{\infty} \int_{0}^{\infty} e^{-\left(x^{2}+y^{2}\right)} d x d y
$$
(c) By changing to polar coordinates, evaluate the double integral in (b), and hence, show that
$$
L\left[t^{-1 / 2}\right]=\sqrt{\frac{\pi}{s}}, s > 0
$$