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 $\begin{array}{l}\rm E_1\end{array}$ .

For the small arguments $\begin{array}{l}|x| \ll 1\end{array}$ it can be expressed as slowly converging sum:

(4)	$\begin{array}{l}\displaystyle {\rm Ei}(x) = \gamma + \ln \|x\| + \sum_{k=1}^{\infty} \frac{x^k}{k \cdot k!}, \quad x>0\end{array}$

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

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Exponential Integral (Ei)