@wikipedia
Two different functions of real argument
are called this way:
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| {\rm Ei}(x) = - \int_{-x}^{\infty} \frac{e^{-\xi}}{\xi} \, d\xi |
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| {\rm E_1}(x) = \int_{x}^{\infty} \frac{e^{-\xi}}{\xi} \, d\xi |
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which are related to each other as:
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{\rm Ei}(x) = - E_1(-x) |
There is a trend to moving from
definition which was dominating in the past towards
.
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Fig. 1. A sample graph of |
Approximations
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| {\rm Ei}(x) = \gamma + \ln |x| + \sum_{k=1}^\infty \frac{x^k}{k\cdot k!} |
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| {\rm Ei}(x) = e^x \, \left[ \frac{1}{x} + \sum_{k=2}^\infty \frac{(k-1)!}{x^k} \right] |
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| {\rm Ei}(-x) \sim \gamma + \ln x |
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| {\rm Ei}(x) \sim \gamma + \ln x |
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Application
The
-based function:
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{\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 planar axial-symmetric diffusion equation:
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| \frac{\partial {\rm w}}{\partial t} = \frac{\partial {\rm w}^2}{\partial^2 r} + \frac{1}{r} \frac{\partial {\rm w}}{\partial r} |
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| {\rm w}(t=0, r) = 0 |
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| {\rm w}(t, r=\infty) = 0 |
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| 0 <= {\rm w}(t, r) < \infty \, , \ \forall (t,r) \in D = \{ t \geq 0, r>0 \}
\subset \mathbb{R} |
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and is widely used in radial mass-heat transfer simulations.
References
https://www.wolframalpha.com/input/?i=Ei(x)
