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$$\int_{-\infty}^{-(-a+\frac{b}{y})^{1/2}}\frac{e^{-x^2}}{x}dx=\tfrac{1}{2}{\rm Ei}\,(a-b/y)=$$ $$\qquad\qquad-\frac{y }{2 b}e^{a-\frac{b}{y}}\left[1-(1-a)y/b+(2-2a+a^2)(y/b)^2\right.$$ $$\qquad\qquad\left.-(6-6a+3a^2-a^3)(y/b)^3+{\cal O}(y/b)^4\right]$$