Exponential Integral (Ei)

Two different functions of real argument $\begin{array}{l}x \in \mathbb{R}\end{array}$ are called this way:

(1)	$\begin{array}{l}\displaystyle {\rm Ei}(x) = - \int_{-x}^{\infty} \frac{e^{-\xi}}{\xi} \, d\xi\end{array}$

(2)	$\begin{array}{l}\displaystyle {\rm E_1}(x) = \int_{x}^{\infty} \frac{e^{-\xi}}{\xi} \, d\xi\end{array}$

which are related to each other as:

(3)	$\begin{array}{l}\displaystyle {\rm Ei}(x) = - E_1(-x)\end{array}$

There is a trend to moving from $\begin{array}{l}\rm Ei\end{array}$ definition which was dominating in the past towards $\begin{array}{l}\rm E_1\end{array}$ .

Approximations:

$\begin{array}{l}|x| \ll 1\end{array}$

$\begin{array}{l}|x| \gg 1\end{array}$

(4)	$\begin{array}{l}\displaystyle {\rm Ei}(x) = \gamma + \ln \|x\| + \sum_{k=1}^\infty \frac{x^k}{k\cdot k!}\end{array}$

(5)	$\begin{array}{l}\displaystyle {\rm Ei}(x) = e^x \, \left[ \frac{1}{x} + \sum_{k=2}^\infty \frac{(k-1)!}{x^k} \right]\end{array}$

$\begin{array}{l}-1 \ll x < 0\end{array}$

$\begin{array}{l}0 < x \ll 1\end{array}$

(6)	$\begin{array}{l}\displaystyle {\rm Ei}(x) = -\gamma - \ln (-x)\end{array}$

(7)	$\begin{array}{l}\displaystyle {\rm Ei}(x) = \gamma + \ln x\end{array}$

where $\begin{array}{l}\gamma = 0.577 ...\end{array}$ is Euler–Mascheroni constant.

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Exponential Integral (Ei)