Logarithmic integral function: Difference between revisions

Revision as of 18:55, 18 November 2008

Revision as of 16:40, 27 January 2009

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where <math>\mathcal{O}</math> refers to [[big O notation]]. The full [[asymptotic expansion]] is

:<math> {\rm li} (x) =\sim \frac{x}{\ln x} \sum_{k=0}^{\infty} \frac{k!}{(\ln x)^k} </math>

or

:<math> \frac{{\rm li} (x)}{x/\ln x} =\sim 1 + \frac{1}{\ln x} + \frac{2}{(\ln x)^2} + \frac{6}{(\ln x)^3} + \cdots. </math>

Note that, as an asymptotic expansion, this series is [[divergent series|not convergent]]: it is a reasonable approximation only if the series is truncated at a finite number of terms, and only large values of ''x'' are employed. This expansion follows directly from the asymptotic expansion for the [[exponential integral]].