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| In [[mathematics]], the '''logarithmic integral function''' or '''integral logarithm''' li(''x'') is a [[special function]]. It is relevant in problems of [[physics]] and has [[number theory|number theoretic]] significance, occurring in the [[prime number theorem]] as an [[Approximation|estimate]] of the number of [[prime number]]s less than a given value.
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| [[Image:Logarithmic integral function.svg|thumb|Logarithmic integral function plot]]
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| ==Integral representation==
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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]]:
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| :<math> {\rm li} (x) = \int_0^x \frac{dt}{\ln t}. \; </math>
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| Here, <math>\ln</math> denotes the [[natural logarithm]]. The function <math>1/\ln(t)</math> has a [[mathematical singularity|singularity]] at ''t'' = 1, and the integral for ''x'' > 1 has to be interpreted as a ''[[Cauchy principal value]]'':
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| :<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>
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| ==Offset logarithmic integral==
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| The '''offset logarithmic integral''' or '''Eulerian logarithmic integral''' is defined as
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| :<math> {\rm Li}(x) = {\rm li}(x) - {\rm li}(2) \, </math>
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| or
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| :<math> {\rm Li} (x) = \int_2^x \frac{dt}{\ln t} \, </math>
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| As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.
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| This function is a very good approximation to the number of prime numbers less than x.
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| ==Series representation==
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| The function li(''x'') is related to the ''[[exponential integral]]'' Ei(''x'') via the equation
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| :<math>\hbox{li}(x)=\hbox{Ei}(\ln x) , \,\!</math>
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| which is valid for ''x'' > 0. This identity provides a series representation of li(''x'') as
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| :<math> {\rm li} (e^u) = \hbox{Ei}(u) =
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| \gamma + \ln |u| + \sum_{n=1}^\infty {u^{n}\over n \cdot n!}
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| \quad \text{ for } u \ne 0 \; , </math>
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| where γ ≈ 0.57721 56649 01532 ... is the [[Euler–Mascheroni gamma constant]]. A more rapidly convergent series due to [[Srinivasa Ramanujan|Ramanujan]] <ref>{{MathWorld | urlname=LogarithmicIntegral | title=Logarithmic Integral}}</ref> is
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| :<math>
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| {\rm li} (x) =
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| \gamma
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| + \ln \ln x
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| + \sqrt{x} \sum_{n=1}^\infty
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| \frac{ (-1)^{n-1} (\ln x)^n} {n! \, 2^{n-1}}
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| \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1} .
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| </math>
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| <!-- cribbed from Mathworld, which cites
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| Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 126–131, 1994.
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| -->
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| ==Special values==
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| 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]].
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| li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151…
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| This is <math>-(\Gamma\left(0,-\ln 2\right) + i\,\pi)</math> where <math>\Gamma\left(a,x\right)</math> is the [[incomplete gamma function]]. It must be understood as the [[Cauchy principal value]] of the function.
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| ==Asymptotic expansion==
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| The asymptotic behavior for ''x'' → ∞ is
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| :<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
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| :<math> {\rm li} (x) \sim \frac{x}{\ln x} \sum_{k=0}^\infty \frac{k!}{(\ln x)^k} </math>
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| or
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| :<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>
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| 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]].
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| ==Number theoretic significance==
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| The logarithmic integral is important in [[number theory]], appearing in estimates of the number of [[prime number]]s less than a given value. For example, the [[prime number theorem]] states that:
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| :<math>\pi(x)\sim\operatorname{Li}(x)</math>
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| where <math>\pi(x)</math> denotes the number of primes smaller than or equal to <math>x</math>.
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| == See also ==
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| * [[Jørgen Pedersen Gram]]
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| * [[Skewes' number]]
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| == References ==
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| <references/>
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| *{{AS ref|5|228}}
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| *{{dlmf|id=6|title=Exponential, Logarithmic, Sine, and Cosine Integrals|first=N. M. |last=Temme}}
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| [[Category:Special hypergeometric functions]]
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| [[Category:Integrals]]
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Andera is what you can call her but she never really liked that name. It's not a common factor but what I like doing is to climb but I don't have the time recently. I've usually loved living in Alaska. Invoicing is what I do for a residing but I've always needed my own business.
Look at my blog; psychic love readings; dev.Microoh.com linked internet page,