@wikipedia

Two different functions are called this way:

{\rm Ei}(x) = - \int_{-x}^{\infty} \frac{e^{-\xi}}{\xi} \, d\xi

{\rm E_1}(x) = \int_{x}^{\infty} \frac{e^{-\xi}}{\xi} \, d\xi

which are related to each other as:

{\rm Ei}(x) = - E_1(-x)

For the positive argument  it can be expressed as slowly converging sum:

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