Please help prove Fourier transform pair

Question

Can any one please help me prove the below Fourier transform pair

$$\frac{1}{\sqrt{|\omega|}} =\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty}e^{-i\omega t}\frac{1}{\sqrt{|t|}}\,dt.$$

Answer 1

$$\begin{align}\frac1{\sqrt{2 \pi}} \int_{-\infty}^{\infty} dt \, |t|^{-1/2} e^{-i \omega t} &= \Re{\left[\sqrt{\frac{2}{\pi}} \int_0^{\infty} dt \, t^{-1/2} e^{-i \omega t}\right]} \\ &= 2 \sqrt{\frac{2}{\pi}} \Re{\left [\int_0^{\infty} du \, e^{-i \omega u^2}\right ]}\\ &= \sqrt{\frac{2}{\pi}} \Re{\left [\sqrt{\frac{\pi}{i \omega}}\right ]}\\ &= |\omega|^{-1/2} \end{align}$$