Logarithmic integral function: Difference between revisions
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{{Short description|Special function defined by an integral}} |
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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. In particular, according to the [[ |
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{{Redirect|Li(x)|the polylogarithm denoted by Li<sub>''s''</sub>(''z'')|Polylogarithm}} |
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{{Use American English|date = January 2019}} |
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[[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 logarithmic integral function li(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D]] |
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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. 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 <math>x</math>. [[Image:Logarithmic integral function.svg|thumb|right|300px|Logarithmic integral function plot]] |
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==Integral representation== |
==Integral representation== |
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The logarithmic integral has an integral representation defined for all positive [[real number]]s {{mvar|x}} ≠ 1 by the [[integral|definite integral]] |
The logarithmic integral has an integral representation defined for all positive [[real number]]s {{mvar|x}} ≠ 1 by the [[integral|definite integral]] |
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:<math> \operatorname{li}(x) = |
:<math> \operatorname{li}(x) = \int_0^x \frac{dt}{\ln t}. </math> |
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Here, {{math|ln}} denotes the [[natural logarithm]]. The function {{math|1/ |
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]], |
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:<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> \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> |
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==Offset logarithmic integral== |
==Offset logarithmic integral== |
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The '''offset logarithmic integral''' or '''Eulerian logarithmic integral''' is defined as |
The '''offset logarithmic integral''' or '''Eulerian logarithmic integral''' is defined as |
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:<math> \operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2) |
:<math> \operatorname{Li}(x) = \int_2^x \frac{dt}{\ln t} = \operatorname{li}(x) - \operatorname{li}(2). </math> |
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or, integrally represented |
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Equivalently, |
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==Special values== |
==Special values== |
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The function li(''x'') has a single positive zero; it occurs at ''x'' ≈ 1.45136 92348 83381 05028 39684 85892 02744 |
The function li(''x'') has a single positive zero; it occurs at ''x'' ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... {{OEIS2C|A070769}}; this number is known as the [[Ramanujan–Soldner constant]]. |
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Li(0) = li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 |
<math>\text{li}(\text{Li}^{-1}(0)) = \text{li}(2)</math> ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... {{OEIS2C|A069284}} |
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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. |
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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\quad \text{ for } u \ne 0 \; , </math> |
\quad \text{ for } u \ne 0 \; , </math> |
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where γ ≈ 0.57721 56649 01532 ... {{OEIS2C|id=A001620}} is the [[Euler–Mascheroni |
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 |
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:<math> |
:<math> |
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\operatorname{li}(x) = |
\operatorname{li}(x) = |
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\gamma |
\gamma |
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+ \ln \ln x |
+ \ln |\ln x| |
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+ \sqrt{x} \sum_{n=1}^\infty |
+ \sqrt{x} \sum_{n=1}^\infty |
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\frac{ (-1)^{n-1} (\ln x)^n} {n! \, 2^{n-1}} |
\left( \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} . |
\sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1} \right). |
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</math> |
</math> |
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<!-- cribbed from Mathworld, which cites |
<!-- cribbed from Mathworld, which cites |
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The asymptotic behavior for ''x'' → ∞ is |
The asymptotic behavior for ''x'' → ∞ is |
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:<math> \operatorname{li}(x) = O \left( {x |
:<math> \operatorname{li}(x) = O \left( \frac{x }{\ln x} \right) . </math> |
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where <math>O</math> is the [[big O notation]]. The full [[asymptotic expansion]] is |
where <math>O</math> is the [[big O notation]]. The full [[asymptotic expansion]] is |
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This gives the following more accurate asymptotic behaviour: |
This gives the following more accurate asymptotic behaviour: |
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:<math> \operatorname{li}(x) - {x |
:<math> \operatorname{li}(x) - \frac{x}{ \ln x} = O \left( \frac{x}{(\ln x)^2} \right) . </math> |
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⚫ | 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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This implies e.g. that we can bracket li as: |
This implies e.g. that we can bracket li as: |
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:<math> 1+\frac{1}{\ln x} < \operatorname{li}(x) |
:<math> 1+\frac{1}{\ln x} < \operatorname{li}(x) \frac{\ln x}{x} < 1+\frac{1}{\ln x}+\frac{3}{(\ln x)^2} </math> |
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for all <math>\ln x \ge 11</math>. |
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==Number theoretic significance== |
==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: |
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{ |
:<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>. |
where <math>\pi(x)</math> denotes the number of primes smaller than or equal to <math>x</math>. |
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Assuming |
Assuming the [[Riemann hypothesis]], we get the even stronger:<ref>Abramowitz and Stegun, p. 230, 5.1.20</ref> |
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:<math>\operatorname{ |
:<math>|\operatorname{li}(x)-\pi(x)| = O(\sqrt{x}\log x)</math> |
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In fact, the [[Riemann hypothesis]] is equivalent to the statement that: |
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:<math>|\operatorname{li}(x)-\pi(x)| = O(x^{1/2+a})</math> for any <math>a>0</math>. |
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For small <math>x</math>, <math>\operatorname{li}(x)>\pi(x)</math> but the difference changes sign an infinite number of times as <math>x</math> increases, and the [[Skewes's number|first time this happens]] is somewhere between 10<sup>19</sup> and 1.4×10<sup>316</sup>. |
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== See also == |
== See also == |
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* [[Jørgen Pedersen Gram]] |
* [[Jørgen Pedersen Gram]] |
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* [[Skewes' number]] |
* [[Skewes' number]] |
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* [[List of integrals of logarithmic functions]] |
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== References == |
== References == |
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{{Reflist}} |
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<references/> |
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*{{AS ref|5|228}} |
*{{AS ref|5|228}} |
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*{{dlmf|id=6|title=Exponential, Logarithmic, Sine, and Cosine Integrals|first=N. M. |last=Temme}} |
*{{dlmf|id=6|title=Exponential, Logarithmic, Sine, and Cosine Integrals|first=N. M. |last=Temme}} |
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{{Nonelementary Integral}} |
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{{Authority control}} |
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[[Category:Special hypergeometric functions]] |
[[Category:Special hypergeometric functions]] |
Latest revision as of 21:51, 7 August 2024
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 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
[edit]The logarithmic integral has an integral representation defined for all positive real numbers x ≠ 1 by the definite integral
Here, ln denotes the natural logarithm. The function 1/(ln t) has a singularity at t = 1, and the integral for x > 1 is interpreted as a Cauchy principal value,
Offset logarithmic integral
[edit]The offset logarithmic integral or Eulerian logarithmic integral is defined as
As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.
Equivalently,
Special values
[edit]The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... OEIS: A070769; this number is known as the Ramanujan–Soldner constant.
≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... OEIS: A069284
This is where is the incomplete gamma function. It must be understood as the Cauchy principal value of the function.
Series representation
[edit]The function li(x) is related to the exponential integral Ei(x) via the equation
which is valid for x > 0. This identity provides a series representation of li(x) as
where γ ≈ 0.57721 56649 01532 ... OEIS: A001620 is the Euler–Mascheroni constant. A more rapidly convergent series by Ramanujan [1] is
Asymptotic expansion
[edit]The asymptotic behavior for x → ∞ is
where is the big O notation. The full asymptotic expansion is
or
This gives the following more accurate asymptotic behaviour:
As an asymptotic expansion, this series is 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.
This implies e.g. that we can bracket li as:
for all .
Number theoretic significance
[edit]The logarithmic integral is important in number theory, appearing in estimates of the number of prime numbers less than a given value. For example, the prime number theorem states that:
where denotes the number of primes smaller than or equal to .
Assuming the Riemann hypothesis, we get the even stronger:[2]
In fact, the Riemann hypothesis is equivalent to the statement that:
- for any .
For small , but the difference changes sign an infinite number of times as increases, and the first time this happens is somewhere between 1019 and 1.4×10316.
See also
[edit]References
[edit]- ^ Weisstein, Eric W. "Logarithmic Integral". MathWorld.
- ^ Abramowitz and Stegun, p. 230, 5.1.20
- Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 5". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 228. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
- Temme, N. M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.