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→Asymptotic expansion: Add another term, to prevent confusion |
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The function li(''x'') is related to the ''[[exponential integral]]'' Ei(''x'') via the equation
:<math>\hbox{li}(x)=\hbox{Ei}(\ln(x)) , \,\!</math>
which is valid for <math>x > 1</math>. This identity provides a series representation of li(''x'') as
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\quad {\rm for} \; u \ne 0 \; , </math>
where γ ≈ 0.57721 56649 01532 ... is the [[Euler-Mascheroni gamma constant]]. A more rapidly convergent series due to [[Ramanujan]] is
:<math>
{\rm li} (x) =
\gamma
+ \ln \ln x
+ \sqrt{x} \sum_{n=1}^{\infty}
\frac{ (-1)^{n-1} (\ln x)^n} {n! \, 2^{n-1}}
\sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1} .
</math>
<!-- cribbed from Mathworld, which cites
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 126-131, 1994.
-->
==Special values==
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