Logarithmic integral function: Difference between revisions

Revision as of 20:55, 20 December 2013

Revision as of 12:34, 3 April 2014

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The '''offset logarithmic integral''' or '''Eulerian logarithmic integral''' is defined as

:<math> {\rm Li}(x) = {\rm li}(x) - {\rm li}(2) \, </math>

or

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==Special values==

The function li(''x'') has a single positive zero; it occurs at ''x'' ≈ 1.45136 92348 ...; this number is known as the [[Ramanujan–Soldner constant]].

li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151…

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==Asymptotic expansion==

The asymptotic behavior for ''x''&nbsp;→&nbsp;∞ is

:<math> {\rm li} (x) = O \left( {x\over \ln x} \right) \; . </math>

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where <math>O</math> is the [[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]].