Exponential Integral (Ei)

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

for the positive argument $\begin{array}{l}x>0\end{array}$ it can be decomposed into converging sum:

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