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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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There is a trend to moving from
definition (which was dominating in the past) towards
which becomes more and more popular nowdays.
 Image Modified
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Fig. 1. A sample graph of |
Properties
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| {\rm Ei}(0) = -\infty |
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| {\rm Ei}(-\infty) = 0 |
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| {\rm Ei}(+\infty) = +\infty |
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| \frac{d }{dx}{\rm Ei}(x) = \frac{e^x}{x} |
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Approximations
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body | | |
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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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-1 \ll -x <0 | |
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| {\rm Ei}(- |
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x) \sim \gamma + \ln | LaTeX Math Block |
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{\rm Ei}(x
Application
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The real-value positive function of two real-value positive arguments (time and radial coordinate ):
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defines a solution of planar axial-symmetric diffusion equation with homogenous initial and boundary conditions:
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| \frac{\partial {\rm w}}{\partial t} = \frac{\partial^2 {\rm w}}{\partial r^2} + \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 heat-mass transfer analysis.
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