Editing Exponential integral (section)

==== Asymptotics beyond all orders ====
Using integration by parts, we can obtain an explicit formula<ref>{{Citation |last=O’Malley |first=Robert E. |title=Asymptotic Approximations |date=2014 |url=https://doi.org/10.1007/978-3-319-11924-3_2 |work=Historical Developments in Singular Perturbations |pages=27–51 |editor-last=O'Malley |editor-first=Robert E. |access-date=2023-05-04 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-11924-3_2 |isbn=978-3-319-11924-3|url-access=subscription }}</ref><math display="block">\operatorname{Ei}(z) = \frac{e^{z}} {z} \left (\sum _{k=0}^{n} \frac{k!} {z^{k}} + e_{n}(z)\right), \quad e_{n}(z) \equiv (n + 1)!\ ze^{-z}\int _{ -\infty }^{z} \frac{e^{t}} {t^{n+2}}\,dt</math>For any fixed <math>z</math>, the absolute value of the error term <math>|e_n(z)|</math> decreases, then increases. The minimum occurs at <math>n\sim |z|</math>, at which point <math>\vert e_{n}(z)\vert \leq \sqrt{\frac{2\pi } {\vert z\vert }}e^{-\vert z\vert }</math>. This bound is said to be "asymptotics beyond all orders".