Define
$$f(x) := \begin{cases} \displaystyle\frac{1}{\sqrt{2\pi}}\frac{e^{-1/2x}}{x^{3/2}}, & \text{for $x>0$} \\[4ex] 0 & \text{otherwise} \end{cases}$$
Then, how do I prove that $$\displaystyle\int_{-\infty}^{\infty} e^{itx}f(x) dx = e^{-\sqrt{-2it}}$$ for all $t\in \mathbb{R}$?
Since $f$ does not have a moment generating function, we cannot apply analytic continuation technique here to prove the above equality. Hence, I think we must directly prove this; but how?