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

For the positive argument $\begin{array}{l}x>0\end{array}$ it can be expressed as slowly converging sum:

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

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