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Two different functions of real argument x \in \mathbb{R} are called this way:

(1) {\rm Ei}(x) = - \int_{-x}^{\infty} \frac{e^{-\xi}}{\xi} \, d\xi
(2) {\rm E_1}(x) = \int_{x}^{\infty} \frac{e^{-\xi}}{\xi} \, d\xi

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 which becomes more and more popular nowdays.


Fig. 1. A sample graph of y ={\rm Ei}(x)


Properties



(4) {\rm Ei}(0) = -\infty
(5) {\rm Ei}(-\infty) = 0
(6) {\rm Ei}(+\infty) = +\infty
(7) \frac{d }{dx}{\rm Ei}(x) = \frac{e^x}{x}


Approximations



|x| \ll 1

|x| \gg 1

(8) {\rm Ei}(x) = \gamma + \ln |x| + \sum_{k=1}^\infty \frac{x^k}{k\cdot k!}
(9) {\rm Ei}(x) = e^x \, \left[ \frac{1}{x} + \sum_{k=2}^\infty \frac{(k-1)!}{x^k} \right]

where \gamma = 0.57721... is Euler–Mascheroni constant



0 < x \ll 1

(10) {\rm Ei}(-x) \sim \gamma + \ln x


Application


The real-value positive function {\rm w}(t,r) of two real-value positive arguments (time t and radial coordinate r):

(11) {\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

(12) \frac{\partial {\rm w}}{\partial t} = \frac{\partial^2 {\rm w}}{\partial r^2} + \frac{1}{r} \frac{\partial {\rm w}}{\partial r}
(13) {\rm w}(t=0, r) = 0
(14) {\rm w}(t, r=\infty) = 0
(15) 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.


See also


Formal science / Mathematics / Calculus


References






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