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 which becomes more and more popular nowdays.

GLOSSARY > Exponential Integral (Ei or E1) > image2019-3-9_9-34-24.png

Fig. 1. A sample graph of

Properties

{\rm Ei}(0)  = -\infty

{\rm Ei}(-\infty)  = 0

{\rm Ei}(+\infty)  = +\infty

\frac{d }{dx}{\rm Ei}(x) = \frac{e^x}{x}

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

The real-value positive function of two real-value positive arguments (time and radial coordinate ):

 {\rm w}(t, r) = E_1 \left( \frac{r^2}{4 t} \right) = -{\rm Ei} \left( - \frac{r^2}{4 t} \right)

honours a planar axial-symmetric diffusion equation with homogenous initial and boundary conditions:

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

{\rm w}(t=0, r) = 0

{\rm w}(t, r=\infty) = 0

0 <= {\rm w}(t, r) < \infty \, , \ \forall (t,r) \in  D = \{ t \geq 0, r>0  \} 
 \subset \mathbb{R}

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