Two different functions of real argument x \in \mathbb{R} are called this way:
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which are related to each other as:
(3) | {\rm Ei}(x) = - E_1(-x) |
There is a trend to moving from \rm Ei definition which was dominating in the past towards \rm E_1.
Fig. 1. A sample graph of y ={\rm Ei}(x) |
Approximations
|x| \ll 1 | |x| \gg 1 | |||||
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where \gamma = 0.57721... is Euler–Mascheroni constant | ||||||
-1 \ll -x <0 | 0 < x \ll 1 | |||||
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Application
The \rm Ei-based function:
(8) | {\rm w}(t, r) = {\rm Ei} \left( - \frac{r^2}{4 \, t} \right) |
honors the planar axial-symmetric diffusion equation:
(9) | \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.
References
https://www.wolframalpha.com/input/?i=Ei(x)