Logarithmic Integral Function - Series Representation

Series Representation

The function li(x) is related to the exponential integral Ei(x) via the equation

which is valid for x > 0. This identity provides a series representation of li(x) as

${\rm li} (e^u) = \hbox{Ei}(u) = \gamma + \ln |u| + \sum_{n=1}^\infty {u^{n}\over n \cdot n!} \quad \text{ for } u \ne 0 \;,$

where γ ≈ 0.57721 56649 01532 ... is the Euler–Mascheroni gamma constant. A more rapidly convergent series due to Ramanujan is

${\rm li} (x) = \gamma + \ln \ln x + \sqrt{x} \sum_{n=1}^\infty \frac{ (-1)^{n-1} (\ln x)^n} {n! \, 2^{n-1}} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1} .$

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