Logarithmic integral function: Difference between revisions

Revision as of 18:41, 21 December 2005

Revision as of 18:46, 21 December 2005

Line 1:

:''See also [[logarithmic integral]] for other senses.''

{{inuse}}

In [[mathematics]], the '''logarithmic integral function''' or '''integral logarithm''' li(''x'') is a [[function (mathematics)|non-elementaryspecial function]] defined for all positive [[real number]]s <math>x\ne1</math> by the [[integral|definite integral]]:

:<math> {\rm li} (x) = \int_{0}^{x} \frac{dt}{\ln (t)} \; . </math>

Line 9:

:<math> {\rm li} (x) = \lim_{\varepsilon \to 0} \left( \int_{0}^{1-\varepsilon} \frac{dt}{\ln (t)} + \int_{1+\varepsilon}^{x} \frac{dt}{\ln (t)} \right) \; . </math>

Sometimes instead of li the [[offset==Offset logarithmic integral]] is used, defined as==

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

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

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

or

:<math> {\rm Li} (x) = \int_{2}^{x} \frac{dt}{\ln t} \, </math>

==Asymptotic expansion==

The growth behavior of this function for ''x'' &rarr; &infin; is

Line 17 ⟶ 25:

(see [[big O notation]]).

has the [[asymptotic expansion]]

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

or

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

Note that, as an asymptotic expansion, this series is [[divergent series|not convergent]].

==Series representation==