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Eric Kvaalen (talk | contribs) The 3D graph doesn't correstpond to the real function between 0 and 1. The series should not take the absolute value of u in the meromorphic case. |
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{{Redirect|Li(x)|the polylogarithm denoted by Li<sub>''s''</sub>(''z'')|Polylogarithm}}
{{Use American English|date = January 2019}}
[[File:Plot of the logarithmic integral function li(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the logarithmic integral function li(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the absolute value of the logarithmic integral function li(z) in the complex plane from -2-2i to 2+2i with colors
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. In particular, according to the [[prime number theorem]], it is a very good [[approximation]] to the [[prime-counting function]], which is defined as the number of [[prime numbers]] less than or equal to a given value
== Integral representation ==
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Here, {{math|ln}} denotes the [[natural logarithm]]. The function {{math|1/(ln ''t'')}} has a [[mathematical singularity|singularity]] at {{math|1=''t'' = 1}}, and the integral for {{math|''x'' > 1}} is interpreted as a [[Cauchy principal value]],
: <math> \operatorname{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>
However, the logarithmic integral can also be taken to be a [[meromorphic]] complex-valued function in the complex ___domain. In this case it is multi-valued with branch points at 0 and 1, and the values between 0 and 1 defined by the above integral are not compatible with the values beyond 1. The complex function is shown in the figure above. The values on the real axis beyond 1 are the same as defined above, but the values between 0 and 1 are offset by iπ so that the absolute value at 0 is π rather than zero. The complex function is also defined (but multi-valued) for numbers with negative real part, but on the negative real axis the values are not real.
== Offset logarithmic integral ==
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which is valid for ''x'' > 0. This identity provides a series representation of li(''x'') as
: <math> \operatorname{li}(e^u) = \hbox{Ei}(u) =
\gamma + \ln |u| + \sum_{n=1}^\infty {u^{n}\over n \cdot n!}
\quad \text{ for } u \ne 0 \, , </math>
where ''γ'' ≈ 0.57721 56649 01532 ... {{OEIS2C|id=A001620}} is the [[Euler–Mascheroni constant]]. A more rapidly convergent series by [[Srinivasa Ramanujan|Ramanujan]] <ref>{{MathWorld | urlname=LogarithmicIntegral | title=Logarithmic Integral}}</ref> is▼
where ''γ'' ≈ 0.57721 56649 01532 ... {{OEIS2C|id=A001620}} is the [[Euler–Mascheroni constant]]. For the complex function the formula is
: <math> \operatorname{li}(e^u) = \hbox{Ei}(u) =
\gamma + \ln u + \sum_{n=1}^\infty {u^{n}\over n \cdot n!}
\quad \text{ for } u \ne 0 \, , </math>
(without taking the absolute value of u).
▲
: <math>
\operatorname{li}(x) =
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Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 126–131, 1994.
-->
Again, for the meromorphic complex function the term <math>\ln|\ln u|</math> must be replaced by <math>\ln\ln u.</math>
== Asymptotic expansion ==
The asymptotic behavior both for
: <math> \operatorname{li}(x) = O \left( \frac{x }{\ln x} \right) . </math>
where <math>O</math> is the [[big O notation]]. The full [[asymptotic expansion]] is
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