Two different functions of real argument 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)

There is a trend to moving from definition which was dominating in the past towards .

Approximations

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

{\rm Ei}(x) = e^x \, \left[ \frac{1}{x} + \sum_{k=2}^\infty \frac{(k-1)!}{x^k} \right]

{\rm Ei}(-x) \sim \gamma + \ln x

{\rm Ei}(x) \sim \gamma + \ln x

Application

The -based function:

 {\rm w}(t, r) = {\rm Ei} \left( - \frac{r^2}{4 \,  t} \right)

\frac{\partial {\rm w}}{\partial t} = \frac{\partial {\rm w}^2}{\partial^2 r} + \frac{1}{r} \frac{\partial {\rm w}}{\partial r}

and is widely used in radial mass-heat transfer simulations.