I have a positive random variable $X>0$. I have to compute the probability density function $$PDF_{X}(x)$$ I can compute in closed-form the extended characteristic function ($z \in \mathbb{C}$) $$ CF_{X}(z) = E\left[ e^{iz X^2} \right] $$ that is analytic in the half-plane $Im(z) < \zeta$ with $\zeta > 0$ (*). Unfortunately, this function decays too slowly $$ \lim_{Re(z) \rightarrow +\infty} Re \left[ e^{-izy}CF_{X}(z) \right] \sim \omicron \left( \left( \frac{1}{Re(z)} \right)^{\epsilon} \right), \epsilon < 1 $$ and therefore the $PDF_{X}$ cannot be retrieved as a Fourier inversion of $CF_{X}(z)$, because the defining integral $$ PDF_{X}(x) = \frac{1}{2 \pi} \int^{i Im(z) + \infty}_{i Im(z) + \infty} e^{-izx}CF_{X}(z) dz = \infty $$ How can I do?
(*) It has a logarithmic branch-cut for $Im(z) \in [\zeta, +\infty)$.
