The blue curve is the desired integral $\int_ 0^{\infty}\frac {e^{-x - \frac {1} {x}}} {x} dx$, the orange curve is the approximate answer $2K_0(2)a^3(1+a^3)^{-1}$.
http://ilorentz.org/beenakker/MO/BesselK.png
Carlo Beenakker
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