# Expansions in terms of a variable in the integration limit in a point where the integration limit is not differentiable

Consider the following integral $$\int_{-\infty}^{-(-a+\frac{b}{y})^{1/2}}\frac{e^{-x^2}}{x}dx$$ where $a,b>0$. The integral can be solved exactly but I am not interested in that. I want to perform the expansion of the integral around $y=0$, maybe using something analogous to Leibnitz's rule for computing derivatives of integrals. I am aware that the dependence on $y$ in the integration limits is not differentiable around $y=0$ so I expect the result to be something like $$e^{\frac{c_0}{y}}(c_1+c_2y+c_3y^2+\ldots)$$ but I have no idea how to approach this. So, how can I tackle this?

$$\int_{-\infty}^{-(-a+\frac{b}{y})^{1/2}}\frac{e^{-x^2}}{x}dx=\tfrac{1}{2}{\rm Ei}\,(a-b/y)=$$ $$\qquad\qquad=\tfrac{1}{2}(a-b/y)^{-1}\exp(a-b/y)\sum_{n=0}^\infty\frac{n!}{(a-b/y)^n}$$ $$\qquad\qquad=-\frac{y }{2 b}e^{a-\frac{b}{y}}\sum_{n=0}^\infty (-1)^{n}n!(1-a)^n(y/b)^n$$