I am trying to obtain an analytic estimate of this integral:


$\int\_0^1\frac{1}{\sqrt{x}}\exp\left(-a(x-x\_0)^2\right) dx$,


where $a\gg1$, $x\_0\in[0,1]$. Saddle-point approximation doesn't work due to infinite derivative of $1/\sqrt{x}$ at 0. Any tips on how to get a handle on this will be much appreciated.



Estimating an integral with a singularity at the interval's endpoint
Well, now you tell us that $x\_0$ depends on $a$. If $x\_0$ is of order $1/\sqrt{a}$, set $x\_0=y\_0/\sqrt{a}$, $x=y/\sqrt{a}$, and your integral becomes
$$a^{-1/4}\int\_0^{\sqrt{a}}{1\over\sqrt{y}}\exp(-(y-y\_0)^2)\,dy.$$
Up to exponentially small error, you can replace the upper bound of $\sqrt{a}$ by $\infty$. 


