Upper bounding the exponential integral $\mathrm{Ei}(x)$

Question

Are there any references out there that deal with upper bounds on the exponential integral: $$ \mathrm{Ei}(x) = -\int_{-x}^{\infty} \frac{e^{-t}}{t} dt= \gamma + \ln{x} + \sum_{n=1}^\infty \frac{x^n}{nn!}$$ where $\gamma$ is Euler's constant. I understand that $e^x + \ln{x} + \gamma$ is a trivial upper bound, but my application requires something tighter. Would also appreciate any advice on how one might tackle such a problem.

Answer 1

Note that for $x > 0$ \begin{align} \operatorname{Ei} (x) - \gamma - \ln (x) &= \sum \limits_{n=1}^\infty \left[\frac{x^n}{n!} - \left(1-\frac{1}{n}\right) \frac{x^n}{n!}\right] \\ &= \mathrm{e}^x - 1 - \sum \limits_{n=2}^\infty \frac{x^n}{n^2 (n-2)!} \\ &= \mathrm{e}^x - 1 - \frac{x^2}{4} - \frac{x^3}{9} - \frac{x^4}{32} - \dots \, . \end{align} You can estimate the left-hand side from above by terminating the series on the right-hand side at an arbitrary order $N \in \mathbb{N}$: $$ \operatorname{Ei} (x) - \gamma - \ln (x) < \mathrm{e}^x - 1 - \sum \limits_{n=2}^N \frac{x^n}{n^2 (n-2)!} \, .$$ This gives an arbitrarily tight bound for small $x > 0$. For sufficiently large $x$, however, the exponential function will dominate the polynomial and the approximation will become worse.

Answer 2

From $\S6.12(\text{i})$ of the DLMF, we can see that $$ \operatorname{Ei}(x) = \frac{{{\rm e}^x }}{x}\left( {\sum\limits_{n = 0}^{N - 1} {\frac{{n!}}{{x^n }}} + R_N (x)} \right) $$ for all $x>0$ and $N =0,1,2,\ldots$, where $$ \left| {R_N (x)} \right| \le \frac{{N!}}{{x^N }} \cdot \left( {1 + \sqrt \pi \frac{{\Gamma\! \left( {\frac{{N + 1}}{2} + 1} \right)}}{{\Gamma\! \left( {\frac{{N + 1}}{2} + \frac{1}{2}} \right)}}} \right). $$ It can be shown that $$ \sqrt \pi \frac{{\Gamma \!\left( {\frac{{N + 1}}{2} + 1} \right)}}{{\Gamma\! \left( {\frac{{N + 1}}{2} + \frac{1}{2}} \right)}} \le \sqrt {\frac{\pi }{2}\left( {N + \frac{\pi }{2}} \right)} $$ for $N =0,1,2,\ldots$.