I am interested in the integral $$I(h,a,b) = \sqrt{h} \int_{-\infty}^{\infty} du \frac{1}{\sqrt{u}} e^{-h u^2 -i(au -{b}/{u})}$$ as $h \to \infty$.
$a,b,h$ are all real parameters. Here the square root is defined for $u < 0$ as $\sqrt{u} = i\sqrt{-u}$.
I expect for $b=0$ the integral will blow up in the limit $h \to \infty$ like $\frac{(1-i)}{2}\Gamma(\frac{1}{4})h^\frac{1}{4}$ via a saddle-point style approximation.
I anticipate that for $b \neq 0$ the limit will be far more precarious (and $I$ may even dwindle to zero(!), which I see from numerics) as $h \to \infty$. I feel a little confused about how to handle behavior around the origin because of the essential singularity there.
For $b>0$, (and $a$ of either sign), what is the behavior of $I(h,a,b)$ as $h \to \infty$? In particular, what are the $a$ and $b$ dependences?