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

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

Read more about this topic: Logarithmic Integral Function

Famous quotes containing the word series:

“In the order of literature, as in others, there is no act that is not the coronation of an infinite series of causes and the source of an infinite series of effects.”
—Jorge Luis Borges (1899–1986)