@wikipedia
Two different functions of real argument
are called this way:
| LaTeX Math Block |
|---|
| {\rm Ei}(x) = - \int_{-x}^{\infty} \frac{e^{-\xi}}{\xi} \, d\xi |
|
| LaTeX Math Block |
|---|
| {\rm E_1}(x) = \int_{x}^{\infty} \frac{e^{-\xi}}{\xi} \, d\xi |
|
which are related to each other as:
| LaTeX Math Block |
|---|
|
{\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.
|
Fig. 1. A sample graph of |
Properties
| LaTeX Math Block |
|---|
| {\rm Ei}(0) = -\infty |
|
| LaTeX Math Block |
|---|
| {\rm Ei}(-\infty) = 0 |
|
| LaTeX Math Block |
|---|
| {\rm Ei}(+\infty) = +\infty |
|
| LaTeX Math Block |
|---|
| \frac{d }{dx}{\rm Ei}(x) = \frac{e^x}{x} |
|
Approximations
| |
|---|
| LaTeX Math Block |
|---|
| {\rm Ei}(x) = \gamma + \ln |x| + \sum_{k=1}^\infty \frac{x^k}{k\cdot k!} |
|
| LaTeX Math Block |
|---|
| {\rm Ei}(x) = e^x \, \left[ \frac{1}{x} + \sum_{k=2}^\infty \frac{(k-1)!}{x^k} \right] |
|
|
|
| |
| LaTeX Math Block |
|---|
| {\rm Ei}(-x) \sim \gamma + \ln x |
|
| LaTeX Math Block |
|---|
| {\rm Ei}(x) \sim \gamma + \ln x |
|
Application
The real-value positive function of two real-value positive arguments (time and radial coordinate ):
| LaTeX Math Block |
|---|
|
{\rm w}(t, r) = E_1 \left( \frac{r^2}{4 t} \right) = -{\rm Ei} \left( - \frac{r^2}{4 t} \right) |
defines a solution of planar axial-symmetric diffusion equation with homogenous initial and boundary conditions:
| LaTeX Math Block |
|---|
| \frac{\partial {\rm w}}{\partial t} = \frac{\partial^2 {\rm w}}{\partial r^2} + \frac{1}{r} \frac{\partial {\rm w}}{\partial r} |
|
| LaTeX Math Block |
|---|
| {\rm w}(t=0, r) = 0 |
|
| LaTeX Math Block |
|---|
| {\rm w}(t, r=\infty) = 0 |
|
| LaTeX Math Block |
|---|
| 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 simulations.
References
https://www.wolframalpha.com/input/?i=Ei(x)
Masina, Enrico. (2017). A review on the Exponential-Integral special function and other strictly related special functions. Lectures from a seminar of Mathematical Physics.
