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There is a trend to moving from 

LaTeX Math Inline
body\rm Ei
 definition which was dominating in the past towards  
LaTeX Math Inline
body\rm E_1
.



Approximations

LaTeX Math Inline
body|x| \ll 1

LaTeX Math Inline
body|x| \gg 1


LaTeX Math Block
anchor2
alignmentleft
{\rm Ei}(x) = \gamma + \ln |x| + \sum_{k=1}^\infty \frac{x^k}{k\cdot k!}



LaTeX Math Block
anchor2
alignmentleft
{\rm Ei}(x) = e^x \, \left[ \frac{1}{x} + \sum_{k=2}^\infty \frac{(k-1)!}{x^k} \right]


where 

LaTeX Math Inline
body\gamma = 0.57721
... is Euler–Mascheroni constant



LaTeX Math Inline
body-1 \ll -x <0

LaTeX Math Inline
body0 < x \ll 1


LaTeX Math Block
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alignmentleft
{\rm Ei}(-x) \sim \gamma + \ln x



LaTeX Math Block
anchor2
alignmentleft
{\rm Ei}(x) \sim \gamma + \ln x