The standard Levy distribution has the PDF:
$$f(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2x}}\frac{1}{x^{3/2}},$$
where $x\geq0$.
My question is how to compute its characteristic function:
$$ \phi(t)=\int_{-\infty}^{\infty} e^{jtx}f(x)\mathrm{d}x. $$
I tried to use the Residue theorem.