By Mathematica I know that
$$\frac{1}{\sqrt{2\pi}}\int^{\infty}_{0}(1-e^{i\theta x})x^{-3/2}e^{-\frac{b^{2}x}{2}}dx = -b+\sqrt{b^{2}-2i\theta}$$
but unfortunately I cannot compute it analytically. Does anyone have an idea how to bite that problem?
I got that integral from the triplet in the Levy-Khintchine formula calculation for inverse Gaussian process in "Introductory Lectures on Fluctuations of Levy Processes with Applications" by A.E. Kyprianou, where author in excercise 1.7 ask to show that $\int^{\infty}_{0}(1-e^{i\theta x})\Pi(dx) = \psi(\theta)$, where $\psi$ is the characteristic exponent and $\Pi$ is Levy measure.
As a hint, author recommends to use identity
$\int^{\infty}_{0}(1-e^{ir})r^{-\alpha -1}dr = - \Gamma(-\alpha)e^{-i\pi\alpha/2}$