TITLE: Fourier Transform of $\frac{1}{\sqrt{|x|}}$
QUESTION [8 upvotes]: I want to find the fourier transform of $\frac{1}{\sqrt{|x|}}$. I checked the table of common fourier transforms in Wikipedia, and I know the answer should be $$\sqrt{\frac{2\pi}{|\omega|}}$$
What I can't find out, however, is why that is the answer.
I tried 
$$ \hat{f}(\omega) = \int_{-\infty}^{\infty} \frac{1}{\sqrt{|x|}} e^{-i\omega x} dx$$
$$ = \int_{0}^{\infty} \frac{1}{\sqrt{x}} e^{-i\omega x} dx + \int_{0}^{\infty} \frac{1}{\sqrt{x}} e^{i\omega x} dx$$
but that just gives me two unsolvable exponential integrals.
I also tried finding the answer through residue calculus, as the function has a single singularity at 0, which yields
$$ \hat{f}(\omega) = 2\pi i \ Res_{z = 0} \frac{e^{-i \omega z}}{\sqrt{|z|}} = 2\pi i \lim_{z \to 0} (e^{-i \omega z}) = 2\pi i$$
What am I doing wrong? Or am I thinking completely in the wrong direction?
Thanks in advance!

REPLY [2 votes]: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
\begin{align}
&\bbox[#ffd,5px]{\int_{-\infty}^{\infty}{\expo{-\ic\omega x} \over \root{\verts{x}}}\,\dd x} =
2\int_{0}^{\infty}{\cos\pars{\verts{\omega}x} \over \root{x}}\,\dd x
\\[5mm] \stackrel{\verts{\omega}x\ \mapsto\ x}{=}\,\,\,&
{2 \over \root{\verts{\omega}}}
\int_{0}^{\infty}{\cos\pars{x} \over \root{x}}\,\dd x
\\[5mm]
\stackrel{\root{x}\ \mapsto\ x}{=}\,\,\,&
{4 \over \root{\verts{\omega}}}\
\underbrace{\int_{0}^{\infty}\cos\pars{x^{2}}\,\dd x}
_{\ds{{\root{2\pi} \over 4}}}
\\ = &\
\bbx{\root{2\pi} \over \root{\verts{\omega}}}
\\ &
\end{align}
The last integral is a
Fresnel Integral.