Logarithmic integral function: Difference between revisions

Revision as of 11:34, 13 June 2009

Revision as of 17:08, 6 October 2009

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The logarithmic integral has an integral representation defined for all positive [[real number]]s <math>x\ne 1</math> by the [[integral|definite integral]]:

:<math> {\rm li} (x) = \int_{0}int_0^{x} \frac{dt}{\ln (t)}. \; . </math>

Here, ln denotes the [[natural logarithm]]. The function 1/ln (''t'') has a [[mathematical singularity|singularity]] at ''t'' = 1, and the integral for ''x'' > 1 has to be interpreted as a ''[[Cauchy principal value]]'':

:<math> {\rm li} (x) = \lim_{\varepsilon \to 0} \left( \int_{0}int_0^{1-\varepsilon} \frac{dt}{\ln (t)} + \int_{1+\varepsilon}^{x} \frac{dt}{\ln (t)} \right). \; . </math>

==Offset logarithmic integral==

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or

:<math> {\rm Li} (x) = \int_{2}int_2^{x} \frac{dt}{\ln t} \, </math>

As such, the integral representation has the advantage of avoiding the singularity in the ___domain of integration.

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:<math>\hbox{li}(x)=\hbox{Ei}(\ln(x)) , \,\!</math>

which is valid for <math>''x ''&nbsp;> &nbsp;1</math>. This identity provides a series representation of li(''x'') as

:<math> {\rm li} (e^{u}) = \hbox{Ei}(u) =

\gamma + \ln u + \sum_{n=1}^{\infty} {u^{n}\over n \cdot n!}

\quad \text{\rm for } \; u \ne 0 \; , </math>

where γ ≈ 0.57721 56649 01532 ... is the [[Euler-MascheroniEuler–Mascheroni gamma constant]]. A more rapidly convergent series due to [[Srinivasa Ramanujan|Ramanujan]] {{Fact|date=July 2007}} is

:<math>

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\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} .

</math>

<!-- cribbed from Mathworld, which cites

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 126-131126–131, 1994.

-->

==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-SoldnerRamanujan–Soldner constant]].

li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151&hellip;

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==Asymptotic expansion==

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

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

where <math>\mathcal{O}</math> refersis tothe [[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